MATHEXCEL 1611 WORKSHEET FIRST REVIEW

Transcription

MATHEXCEL 1611 WORKSHEET FIRST REVIEW
Calculators are NOT allowed!
1. Show that the following statements are true using two different methods.
a) 27 · 210 = 217
b)
27
= 2−3
10
2
2
c) (23 ) = 23·2
d) (2 · 3)3 = 23 · 33
2. Explain why 2a · 2b = 2a+b .
3. What are the values of 92 , 9−2 , 91/2 ? Do you see a pattern? Can you predict the value of
9−1/2 ? What is the value of 9−1/2 ?
4. Factor the following: a) x2 − 5x − 6
b) x2 − 9
c) x2 − 5
d) x3 − 5x2 − 6x
e) x4 − 16
f) x4 + 24x2 + 144. (Do not use complex numbers.)
√
5. Factor x − 5 into terms involving x.
6. Rewrite the following algebraic expression in one other way, simplifying as much as possible
(a)
x+2
x−2
2
x2 −4
2
x − 3x + 2
x−2
√
4+x−2
(c)
x
1
−1
(d) 3+x 3
x
1
√
− 13
9−x
(e)
x
(b)
MATHEXCEL 1611 WORKSHEET SECTION 1.1
√
1. Find the domain and range of (a) f (x) = 2 − x
(b) f (x) = √
2
9 − x2
(c) f (x) =
2. Graph the following equations and explain why they are not graphs of functions of x.
|x| + |y| = 1
b) |x + y| = 1
x
1−x
a)
3. Graph the functions
(
(a) f (x) =
(
1 − x, 0 ≤ x ≤ 1
x2 , 1 < x ≤ 2
(b) f (x) =
1
,
x
x<0
2x + 1, x ≥ 0
Graph the functions in exercises 4-7. What symmetries, if any, do the graphs have? Specify the
intervals over which the function is increasing and intervals where it is decreasing.
4. y = −
1
x2
5. y =
1
|x|
√
6. y = −4 x
7. y = −x2/3
In Exercises 8-11, state whether the function is even, odd, or neither. Give reasons for your
answers.
8. f (x) = x2 + 4x
9. f (x) = x4 + 3x2 − 1
10. f (x) =
x2
x
−1
11. f (x) = |x3 |
1. Find the domain and ranges of f , g, f /g, and g/f where f (x) = 1 and g(x) = 1 +
2. If f (x) = x − 1 and g(x) =
µ µ ¶¶
√
x.
1
, find the following
x+1
µ µ ¶¶
1
(a) f g
2
(e) f (f (2))
3. If f (x) =
√
1
2
(f) g(g(2))
(b) g f
x, g(x) =
(c) f (g(x))
(d) g(f (x))
(g) f (f (x))
(h) g(g(x))
x
, and h(x) = 4x − 8, find the following
4
(a) h(g(f (x)))
(b) h(f (g(x)))
(c) g(h(f (x)))
(d) g(f (h(x)))
(e) f (g(h(x)))
(f) f (h(g(x))
4. If y = f (x) is the graph, sketch
(d) y = f (x) − 4
(a) y = f (x + 2)
(e) y = f (4x)
(f) y = 2f (x)
(b) y = f (x − 3)
(g) y = f (−x)
(c) y = f (x) + 1
(h) y = −f (x).
y
x
In exercises 5-8 sketch both the original graph and the shifted graph, and give an equation for
the shifted graph.
5. x2 + y 2 = 25, Up 3, left 4
6. y = x2/3 , Right 1, down 1
1
7. y = (x + 1) + 5, Down 5, right 1
2
8. y =
1
, Left 2, down 1
x2
In exercises 9-13 graph the functions.
9. y =
√
9−x
10. y = 1 −
√
x
11. y + 4 = x2/3
12. y =
1
x+2
13. y =
1
(x + 1)2
In exercises 14-16 sketch both the original graph and the stretched/compressed graph, and give
an equation for the stretched/compressed graph.
14. y = x2 − 1, compressed horizontally by a factor of 2.
15. y = 1 +
16. y =
√
1
, stretched horizontally by a factor of 3.
x2
x + 1, stretched vertically by a factor of 3.
1. Evaluate the following:
sin(0)
sin(π/6)
sin(π/4)
sin(π/3)
sin(π/2)
cos(0)
cos(π/6)
cos(π/4)
cos(π/3)
cos(π/2)
tan(0)
tan(π/6)
tan(π/4)
tan(π/3)
tan(π/2)
sin(3π/4)
sin(−3π/4)
sin(7π/4)
sin(−7π/4)
cos(5π/6)
cos(−5π/6)
cos(11π/6)
cos(−11π/6)
cos−1 (0)
cos−1 (1)
cos−1 (−1)
sin−1 (0)
sin−1 (1)
sin−1 (−1)
tan−1 (0)
tan−1 (1)
tan−1 (−1)
√
cos−1 ( 2/2)
√
sin−1 ( 2/2)
√
tan−1 ( 3)
cos−1 (1/2)
cos−1 (−1/2)
sin−1 (1/2)
√
tan−1 (− 3)
sin−1 (−1/2)
4. Using the unit circle, explain the identity sin2 x + cos2 x = 1.
5. Using basic facts from trigonometry, explain the identities tan2 x + 1 = sec2 x and 1 + cot2 x =
csc2 x.
6. sin(−x) =?
cos(−x) =?
7. Given that sin(x + y) = sin x cos y + sin y cos x find an identity for sin(x − y).
8. Given that cos(x + y) = cos x cos y − sin x sin y find an identity for cos(x − y).
9. Show that cos(2x) = cos2 (x) − sin2 (x) = 2 cos2 (x) − 1 = 1 − 2 sin2 (x).
10. Show that sin2 (x) =
1 − cos(2x)
1 + cos(2x)
and that cos2 (x) =
.
2
2
11. Using the identities in Exercise 7 show that sin(nx) cos(mx) = 21 (sin((n+m)x)+sin((n−m)x).
12. Using the identities in Exercise 8 show that cos(nx) cos(mx) = 12 (cos((n+m)x)+cos((n−m)x).
13. Using the identities in Exercise 8 show that sin(nx) sin(mx) = 12 (cos((n−m)x)−cos(n+m)x).
MATHEXCEL 1611 WORKSHEET SECTIONS 1.4-5
a) e0
1. Find the value of
b) e1
c) ln 1
d) ln 0
e) ln e.
Ã
2. Find the exact value of
a) e
2 ln 3
!
e2
c) ln 5 .
e
b) log10 25 + log10 4
3. Rewrite the equation log3 81 = 4 as an exponential statement.
4. Rewrite the equation 43 = 64 as a logarithmic statement.
5. Solve 2 + ln(x + 1) = ln(3x + 2) for x.
6. Solve 2x+1 = 34x−3 for x.
7. Solve e2x − 5ex + 6 = 0 for x.
8. Solve (ln x)2 − ln x − 6 = 0 for x.
2 +3x+2+ln 4
9. Solve ex
= 4 for x.
10. Express 3 ln(x + 2) −
³
´
³
´
1
ln(x − 1) + 2 ln x2 + 3 − 4 ln x2 + 2x + 7 as a single logarithm.
2
!
√
x3/2 x + 1
11. Rewrite ln
as a sum and/or difference of logarithms using ALL of the
(x2 + 2) (x + 3)5
properties of logarithms.
Ã
12. The population of a certain species in a limited environment with initial population 100 and
carrying capacity 1000 is given by
P (t) =
100,000
100 + 900e−t
where t is measured in years. Find the time required for the population to reach 900.
13. Show that the function f (x) =
x+1
is one-to-one.
2x + 1
x+1
. What is the domain of f (x)? What is the domain
2x + 1
−1
of f (x)? What is the range of f (x)? What is the range of f −1 (x)?
14. Find the inverse function for f (x) =
15. Graph
a) y = ex
16. Graph
a) y = ln x
e) y = ln(x + 1).
b) y = e−x
c) y = −ex
b) y = ln(−x)
d) y = ex + 1
c) y = − ln x
e) y = ex+1 .
d) y = ln(x) + 1
In exercises 1-6 find the average value of the function over the given intervals.
1. f (x) = x2
(a) [−2, 2]
(b) [−2, −1]
2. f (x) = sin x
(a) [0, π/2]
√
3. f (x) = x + 4
(a) [0, 5]
4. f (x) = ex
(a) [0, 2]
5. f (x) = ln x
(a) [1, 4]
6. f (x) = x3 − x2 + 3
(c) [1, 5]
(b) [π/6, π/3]
(b) [1, 12]
(c) [π/4, π]
(c) [−4, 5]
(b) [−1, 1]
(c) [1, 3]
(b) [1, e]
(c) 1, e2
(a) [0, 2]
h
(b) [1, 3]
i
(c) [−1, 2]
In exercises 7-12 find (a) the slope of the curve at the given point and (b) the equation of the
tangent line at that point.
7. y = x2 − 3x, P (1, −2)
8. y = x3 − 5x, P (2, −2)
µ
1
1
9. y = , P 2,
x
2
¶
10. y = x2 − 4, P (3, 5)
11. y = 7 − x2 , P (2, 3)
12. y = x2 − 3x + 5, P (2, 3)
x4 − 16
x→2 x − 2
1. Evaluate lim
(2 + x)3 − 8
x→0
x
2. Evaluate lim
1
2
− 2
x→1 x − 1
x −1
x
4. Evaluate lim √
x→0
1 + 3x − 1
√
x − x2
√
5. Evaluate lim
x→1 1 −
x
√
6−x−2
6. Evaluate lim √
x→2
3−x−1
3. Evaluate lim
7. Evaluate
a) lim+
x→0
1
1
−
x |x|
8.
b) lim−
x→0
1
1
−
x |x|



x
h(x) =  x2

8−x
Find
a) lim+ h(x)
b) lim− h(x)
x→0
f) lim h(x)
x→0
c) lim h(x)
x→0
g) lim h(x)
x→2
x→1
9. If 2x − 1 ≤ f (x) ≤ x2 for 0 < x < 3 find lim f (x).
x→1
µ
10. Show that lim x2 cos
x→0
1
x2
¶
= 0.
if x < 0
if 0 ≤ x ≤ 2
if x > 2
d) lim+ h(x)
x→2
e) lim− h(x)
x→2
Evaluate the following limits.
1. a) x→∞
lim
2 + 3x − 4x3
x3 + 1
2 + 3x − 4x3
x→−∞
x3 + 1
b) lim
7x5 − 4x3 + 1
x→∞
6x7 + 2x
7x5 − 4x3 + 1
x→−∞
6x7 + 2x
2. 2. a) lim
3. a) x→∞
lim
6. x→∞
lim
√
3x2 + 8x + 6 −
2
√
11. lim
x→∞
b) lim e−x
x→∞
x
x→0 tan x
x→1
³
20. a) lim+
x→2
b) lim x −
√
x→−∞
x2 + 2x
x→−∞
´
6x2 + 5x
x→∞ (1 − x)(2x − 3)
√
x2 + 4x
14. lim
x→−∞ 4x + 1
12. lim
√
´
x2 − 1
x
x→0 sin 3x
x
b) lim−
x→1 x + 1
x−2
x+1
x2 + 2x
b) lim e−x
x→∞
16. lim
x
x+1
s
8. a) lim e−x
10. x→∞
lim tan−1 x4 − x2
x2 + 1 −
15. lim
19. a) lim+
2
x→−∞
2x2 − 1
x + 8x2
³√
√
3x2 + 3x + 1
sin(x) − x
x
s
5. a) x→∞
lim x −
x→−∞
x→∞
13. lim
b) lim
b) lim cos x
7. a) lim e−x
9. x→∞
lim
1 − 6x5 + x4
x→−∞
x3 + 7x2
1 − 6x5 + x4
x3 + 7x2
4. a) x→∞
lim cos x
b) lim
s
b) lim−
x→2
sin 3x
x→0 sin 2x
17. lim
sin 3x cot 5x
x→0
x cot 4x
18. lim
x−2
x+2
an xn + an−1 xn−1 + · · · + a1 x + a0
, find the limits when
m−1 + · · · + b x + b
x→∞ b xm + b
m
m−1 x
1
0
21. Given the limit lim
a) n > m
b) n = m
c) n < m. Explain your answers.
22. How do the answers in the previous problem change when the limit has x → −∞.
MATHEXCEL 1611 WORKSHEET SECTIONS 2.5
1. Evaluate
2. Evaluate
3. Evaluate
4. Evaluate
1
x→2 x − 2
x
c) lim
x→2 x − 2
1
x→2 x − 2
x
b) lim−
x→2 x − 2
1
x→2 x − 2
x
a) lim+
x→2 x − 2
a) lim+
b) lim−
1
x→−3 x + 3
x
a) lim +
x→−3 x + 3
c) lim
1
x→−3 x + 3
x
b) lim −
x→−3 x + 3
a) lim +
1
x→−3 x + 3
x
c) lim
x→−3 x + 3
b) lim −
c) lim
5. Evaluate
a) lim+ cot x
b) lim− cot x
c) lim cot x
6. Evaluate
a) lim+ csc x
b) lim− csc x
c) lim csc x
7. Evaluate
a) lim+ ln(x − 5)
8. Evaluate
a) lim+
9. Evaluate
a) lim+
x
(x − 5)2
10. Evaluate
a) lim+
x2/3
11. Evaluate
a) lim +
x→0
x→0
x→0
x→0
x→5
√
x→0
x→5
x→0
x
2
x→−1
x→0
b) lim− ln(x − 5)
c) lim ln(x − 5)
x→5
x→5
b) lim−
√
x→0
x
c) lim
x→5
b) lim−
x→0
√
x→0
b) lim−
x3
(x + 1)5
x→0
x
x
(x − 5)2
2
x2/3
b) lim −
x→−1
x
x→5 (x − 5)2
c) lim
c) lim
x→0
x3
(x + 1)5
2
x2/3
x3
x→−1 (x + 1)5
c) lim
Graph the rational functions. Include the graphs and equations of the asymptotes.
12. y =
1
x+1
13. y =
2x
x+1
14. y =
x2 + 1
x−1
15. y =
x2 − 1
2x + 4
Use the following graph of f (x) for Exercises 1–6.
c
¢A
¢ A
¢
A
¢ s A
2
1
−1s
¢
A
¢
¢
0
A
s¢
A
Ac
1
s
2
3
&c−1
1. a) Does f (−1) exist?
b) Does limx→−1+ f (x) exist?
d) Is f continuous at x = −1?
b) Does limx→1 f (x) exist?
2. a) Does f (1) exist?
d) Is f continuous at x = 1?
3. a) Is f defined at x = 2? Why or why not?
not?
c) Does limx→−1+ f (x) = f (−1)?
c) Does limx→1 f (x) = f (1)?
b) Is f continuous at x = 2? Why or why
4. At what values of x is f continuous?
5. a) What is the value of limx→2 f (x)?
b) What value should be assigned to f (2) to make f continuous at x = 2?
6. To what new value should f (1) be changed to make f continuous at x = 1?
7. Find c to make the following function continuous and sketch the function.
(
f (x) =
x2 − c2 if x < 4
cx + 20 if x ≥ 4
8. Find a and b to make the following function continuous:
f (x) =


 a+x
2


x
b + 3x
if x < −2
if − 2 ≤ x ≤ 2
if x > 2
9. Use the Intermediate Value Theorem to show that x5 − x2 − 4 = 0 has at least one solution.
See other side for remaining exercises.
10. Use the Intermediate Value Theorem to show that ex = 2 − x has at least one solution.
11. Sketch an example of a graph of a function that is continuous on the interval [−2, 4] and is not
continuous at x = 5 and at x = −4. Explain how you represented continuity and discontinuity
on your graph.
Find a) lim+ f (x)
x→0
b) lim− f (x)
x→0
c) lim f (x).
x→0
12. Sketch an example of a graph of a function that is continuous on the interval [−4, 6] and
contains the points (1, 4) and (5, 1). Use the Intermediate Value Theorem and your graph to
find a value c such that f (c) = 2. Explain why such a value of c must exist.
MATHEXCEL 1611 SECTION 2.7
1
at the point where x = a.
5 − 2x
µ
¶
1
(b) Find equations of the tangent lines at the points (2, 1) and at −2,
.
3
1. (a) Find the slope of the tangent to the curve y = √
2. (a) Find the slope of the tangent line to the curve y = x3 at the point (−1, −1).
(b) Find the equation of the tangent line in part (a).
1
3. Find an equation of the tangent line to the curve y = √ , at (1, 1).
x
4. Find an equation of the tangent line to the curve y =
x
, at (0, 0).
1−x
t2
5. The displacement in meters of an object moving in a straight line is given by s = 1 + 2t + ,
4
where t is measured in seconds.
(a) Find the average velocity over the following time periods,
i. [1, 3]
ii. [1, 2]
iii. [1, 1.5]
iv. [1, 1.1]
(b) Find the instantaneous velocity when t = 1.
6. The displacement in meters of an object moving in a straight line is given by s = t2 − 8t + 18,
where t is measured in seconds.
(a) Find the average velocity over the following time periods,
i. [3, 4]
ii. [3.5, 4]
iii. [4, 5]
iv. [4, 4.5]
(b) Find the instantaneous velocity when t = 4.
7. A particle moves according to a law of motion s = f (t) = t2 − 10t + 12, for t ≥ 0, where t is
measured in seconds and s in feet.
(a) Find the velocity at time t.
(b) What is the velocity after 3 seconds?
(c) When is the particle at rest?
8. If an arrow is shot upward on the moon with a velocity of 58 m/s, its height after t seconds
is given by H = 58t − 0.8t2 .
(a) Find the velocity of the arrow after one second.
(b) Find the velocity of the arrow when t = a.
(c) When will the arrow hit the moon?
(d) With what velocity will the arrow hit the moon?
MATHEXCEL 1611 WORKSHEET SECTIONS 3.1
1. Find the derivative of f (x) =
√
2x + 3 at x = 3.
2. Find the equation of the line tangent to f (x) =
line normal to f (x) at x = 3.
√
3. Find the derivative of f (x) = 2x + 3.
2x + 3 at x = 3. Find the equation of the
1
at x = 4.
x+1
normal to f (x) at x = 4.
√
1
at x = 4. Find the equation of the line
x+1
1
.
x+1
1
7. Find the derivative of f (x) = √ at x = 9.
x
1
8. Find the equation of the line tangent to f (x) = √ at x = 9. Find the equation of the line
x
normal to f (x) at x = 9.
1
9. Find the derivative of f (x) = √ .
x
1
at x = −1.
x2
normal to f (x) at x = −1.
1
at x = −1. Find the equation of the line
x2
1
.
x2
13. Find the equations for the lines containing the point (1, 0) and tangent to the curve f (x) = x2 .
(Note: (1, 0) is not on the curve.)
14. Find the equations for the lines containing the point (2, 1) and tangent to the curve f (x) = x2 .
(Note: (2, 1) is not on the curve.)
15. Find the derivative of f (x) = √
1
at x = 3.
x+1
16. Find the equation of the line tangent to f (x) = √
line normal to f (x) at x = 3.
17. Find the derivative of f (x) = √
1
.
x+1
1
at x = 3. Find the equation of the
x+1
In exercises 1–10 find y 0 .
1. y =
√
x(x − 1)
³
4. y = x3 + 1
7. y = x
9. y =
4/3
´2
−x
2/3
³
´
2. y = ex x2 − 2x .
5. y = ex+3 + 2
3. y =
6. y =
√
√
3
√
x2 + 2 x3
1
x− √
x
ex (x2 + 3x)
8. y =
x2 − 1
(1 + ex ) (x2 − 2x + 6)
(x − 1) (ex + ex+3 )
10. y = xr ex . (Simplify completely by factoring.)
In exercises 11–13 find the derivative without using the quotient rule!
11. y =
x3 + 2x
x
12. y =
x3 + 2x
√
x
13. y =
x2 − 3x + 2
x−1
In exercises 14–15 find the derivative using the extended product rule!
14. y = (x − 1)(x + 1)(x − 2)(x + 2)
15. y = (x − 1)(x + 2)(x − 3)(x + 4)
16. Find the equation of the line tangent to y = x2 + 2ex at the point (0, 2). Find the equation of
the normal line at the point (0, 2).
17. For what values of x does the graph of f (x) = 2x3 − 3x2 − 6x + 87 have a horizontal tangent?
18. At what point on the curve y = 1 + 2ex − 3x is the tangent line parallel to the line 3x − y = 5?
x−1
19. Find the equations of the lines tangent to the curve y =
that are parallel to the line
x+1
x − 2y = 2.
20. Let
(
f (x) =
x2
mx + b
if x ≤ 2
if x > 2
Find the values of m and b that make f differentiable.
Calculators are NOT allowed (except for exercise 10 (c))!
1. The position of a body at time t sec is s = t3 − 6t2 + 9t m. Find the body’s acceleration each
time the velocity is zero.
2. The velocity of a body at time t sec is v = 2t3 − 9t2 + 12t − 5 m/sec. Find the body’s speed
each time the acceleration is zero.
³
´
3. The function C(t) = K e−at − e−bt , where a, b, and K are positive constants and b > a, is
used to model the concentration at time t of a drug injected into the bloodstream.
(a) Show that lim C(t) = 0.
t→∞
(b) Find C 0 (t), the rate at which the drug is cleared from circulation.
(c) When is this rate equal to 0?
√
4. A particle moves along a horizontal line so that its coordinate at time t is x = b2 + c2 t2 , t ≥ 0,
where b and c are positive constants.
(a) Find the velocity and acceleration functions.
(b) Show that the particle always moves in the positive direction.
5. A particle moves on a vertical line so that its coordinate at time t is y = t3 − 12t + 3, t ≥ 0.
(a) Find the velocity and acceleration functions.
(b) When is the particle moving upward and when is it moving downward?
(c) Find the distance that the particle travels in the time interval 0 ≤ t ≤ 3.
6. The volume of a right circular cone is V = πr2 h/3, where r is the radius of the base and h is
the height.
(a) Find the rate of change of the volume with respect to the height if the radius is constant.
(b) Find the rate of change of the volume with respect to the radius if the height is constant.
³
√ ´
7. The mass of part of a wire is x 1 + x kilograms, where x is measured in meters from one
end of the wire. Find the linear density of the wire when x = 4 m.
8. A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of
60 cm/sec. Find the rate at which the area within the circle is increasing after (a) 1 sec, (b)
3 sec, and (c) 5 sec. What can you conclude?
9. If p(x) is the total value of the production when there are x workers in a plant, then the
p(x)
average productivity of the work force at the plant is A(x) =
.
x
0
(a) Find A (x). Why does the company want to hire more workers if A0 (x) > 0?
(b) Show that A0 (x) > 0 if p0 (x) is greater than the average productivity.
10. Suppose that the cost, in dollars, for a company to produce x pairs of a new line of jeans is
C(x) = 2000 + 3x + 0.01x2 + 0.0002x3 .
(a) Find the marginal cost function.
(b) Find C 0 (100) and explain its meaning. What does it predict.
(c) Compare C 0 (100) with the cost of manufacturing the 101st pair.
1. In terms of adjacent, opposite, and hypotenuse, write down the formulas for sin θ, cos θ, tan θ,
csc θ, sec θ, and cot θ.
π π π π 2π 3π 5π
7π 5π
2. Find the values of sin x, cos x, and tan x for x = 0, , , , ,
,
,
, π,
,
,
6 4 3 2 3
4
6
6
4
4π 3π 5π 7π 11π
,
,
,
,
, and 2π.
3
2
3
4
6
3. Write down the formulas (and memorize them) for: sin(x + y)
cos(x − y)
sin(2x)
cos(2x)
4. Answer the Pythagorean identities:
1 + cot2 x =?.
sin2 x + cos2 x =?,
sin(x − y)
cos(x + y)
tan2 x + 1 =?, and
1 + cos(2x)
1 − cos(2x)
and sin2 x =
.
2
2
(Hint: use the cos(2x) formula and the identity sin2 x + cos2 x = 1.)
√
6. Find the cos(Arctan 3/4), csc(Arcsec 2), and sin(Arccos x) (for x > 0).
5. From the identities in exercises 3 and 4 show that cos2 x =
In exercises 7–14 find the derivative.
7. f (x) = x2 tan x
10. f (x) =
(x3
8. f (x) =
cos(x) + 1
+ 3x − 4) sin x
13. f (x) = cos2 x − sin2 x
cot x
sin(x) + x
9. f (x) =
11. f (x) = x sin(x) cos(x)
14.
ex tan x
(1 + x3 ) (1 + sin x)
15. Find the line tangent to f (x) = sec x + tan x at x =
16. Find the line tangent to f (x) =
µ ¶
17. Evaluate lim x sin
x→∞
1
.
x
sin 10x
.
x→0 sin 7x
18. Evaluate lim
sin(x − 1)
.
x→1 x2 + x − 2
19. Evaluate lim
1 − cos x
.
x→0
x
20. Evaluate lim
π
.
4
sin x
5π
at x =
.
1 + tan x
4
sec x
+ 2x
x3
12. f (x) = sin2 x + cos2 x
MATHEXCEL 1611 WORKSHEET SECTION 3.5 SHEET 1
2
1. f (x) = (ex )2
2. f (x) = ex
3. f (x) = esin x
4. f (x) = sin (ex )
5. f (x) = tan
7. f (x) = 23
³√
x2 + 1
´
6. f (x) = 24x
x
Ã
ecsc x
9. f (x) = 2
x +1
10. f (x) =
sin (x2 + 2x + 3)
csc (x2 + 2x + 3)
11. f (x) =
13. f (x) = x − 3x
15. f (x) =
−5/4
x3 + 2
cos 2
x +1
tan3 (x5 + 5x)
cot4 (x + 3)
s
17. f (x) =
´
r
x+
!
14. f (x) = cos
µ³
!
2x3 + 5x2 + 6x + 1
h³
x1/2 − x3 + 1
´³
´4 ¶
3x3 + 2x−5/2 + π
´i
!4
16. f (x) = sin (sin (sin (sin x)))
q
x+
e2x
e3x + 1
12. f (x) = csc3
Ã
2
Ã
³
8. f (x) = tan3 x2 + 2x − 1
x+
√
x
³
18. f (x) = tan3 x2 + 1 +
√
³
x3 + 3x + csc x3 − 2x
´´
19. Find the equation of the tangent line to y = sin x + cos(2x) at (π/6, 1).
20. Find the equation of the tangent line to y = 10x at (0, 1).
21. Find the x–coordinates of all the points from 0 ≤ x ≤ 2π on the curve y = 2 sin x − sin(2x)
at which the tangent line is horizontal.
In the following exercises, sketch the curve by using the parametric equations to plot points.
Indicate with an arrow the direction in which the curve is traced as t increases. Eliminate the
parameter to find a Cartesian equation of the curve.
22. x = cos 2t,
y = sin 2t,
24. x = 4 sin t,
y = 2 cos t,
26. x = 3t,
y = 9t2 ,
28. x = sec2 t − 1,
0≤t≤π
0≤t≤π
−∞ < t < ∞
y = tan t,
−
2. x = sin 2πt,
y = cos 2πt,
4. x = 4 cos t, y = 5 sin t,
√
6. x = − t, y = t, t ≥ 0
π
π
≤t≤
2
2
8. x = csc t,
0≤t≤1
0≤t≤π
y = cot x,
0<t<π
29. Find an equation of the tangent to the curve x = t sin t, y = t cos t at t = π.
30. Find
dy
d2 y
and 2 .
dx
dx
(a) x = t3 + t2 + 1, y = 1 − t2
2
(c) x = 1 + t , y = t ln t
(b) x = 1 + tan t, y = cos 2t
(d) x = t cos t,
y = t sin t
31. Find the slope of the tangent line to the given curve at the point corresponding to the specified
value of the parameter.
(a) x = ln t,
y = 1 + t2 ;
(c) x = t − sin t,
t=1
y = 1 + sin t;
t=
π
2
√
(b) x = tet , y = 1 + 1 + t; t = 0
√
√
(d) x = − t + 1, y = 3t; t = 3
32. Show that the curve x = cos t, y = sin t cos t has two tangents at (0, 0) and find their
equations.
33. At what point does the curve x = 1 − 2 cos2 t, y = (tan t) (1 − 2 cos2 t) cross itself? Find the
equations of both tangents at that point.
34. Find equations of the tangents to the curve x = 3t2 + 1,
point (4, 3).
y = 2t3 + 1 that pass through the
35. At what points does the curve
x = 2a cos t − a cos 2t
have vertical or horizontal tangents?
y = 2a sin t − a sin t
MATHEXCEL 1611 WORKSHEET SECTION 3.5 SHEET 2
3
1. Determine where f (x) = (x2 + 2x) has a horizontal tangent line.
µ
2. Determine where f (x) =
Ã
3. Determine where f (x) =
3x + 1
x−2
x2
6−x
¶1/2
has a horizontal tangent line.
!3
has a horizontal tangent line.
Ã
4. Find an equation for the line tangent to f (x) =
Ã
5. Find an equation for the line tangent to f (x) =
3x3 + 6x
2x + 1
2x2 + 2x
5x + 1
Ã
6. Find an equation for the line tangent to f (x) = sin
2
at x = 0.
!3/2
at x = 1.
π (x2 + x)
3
2 +2x+1
7. Find an equation for the line tangent to f (x) = ex
!3
!
at x = 1.
at x = −1.
8. Find an equation for the line tangent to f (x) = 10x at x = 1.
f (x)
can be written as the product f (x) (g(x))−1 . So, instead of using
g(x)
the quotient rule, we could use the product rule and chain rule. Find the derivative of the following
quotients using this method.
Notice that any quotient
9. y =
x2 + 1
x2 − 3
10. y =
2x + 1
3x − 2
11. y =
sin x
ex
1. x3 y + xy − xy 3 = 0
2.
´−1/2
x ³
+ 1 + x2 y 3
= x4
y
Ã
3. x =
x+y
x−y
!2
4. x2 y 1/2 − y 3/2 = 7
5. ex sin y + ey cos x = 2x2
√
e2x 1 + y 2 + sin x
6.
= tan(3y)
x cos y
7. Find the lines tangent and normal to xy + y 2 = x3 − 1 at (1, 0).
8. Find the lines tangent and normal to x3 + xy + y 3 = 11 at (1, 2).
√
9. Find the lines tangent and normal to x2 + y 2 − 5 x2 + y 2 = 0 at (4, 3).
10. Find the lines tangent and normal to x2 + 2xy + y 2 + 2x − y = 26 at (2, 3).
11. Find the points on x2 + 2y 2 = 1 where the tangent line has slope 1.
12. Find the points on x2 y 2 + xy = 2 where the tangent line has slope −1.
13. Find the lines normal to the curve xy + 2x − y = 0 that are parallel to the line 2x + y = 0.
14. Find the equationes of both tangent lines to the ellipse x2 + 4y 2 = 36 that pass through the
point (12, 3). Note: the point (12, 3) does not lie on the ellipse.
Ã
(x + 1) sin x
1. y = ln √ 2
x + 2 (1 + ex )
³
2. y = ln x2 ex
!
´
3. y = log2 (sin x)
4. y = ln(sec x + tan x)
s
5. y = ln
4
Simplify your answer.
x2 + 1
x−1
6. y = ln (ln (ln x))
7. y = (sin x)tan x
8. y = (ln x)x
9. Find y 0 if xy = y x .
10. Find y (4) if y = x3 ln x.
11. Use logarithmic differentiation to find the derivative of y =
12. Use logarithmic differentiation to find the derivative of y =
³
√
xex
2
³
x2 + 1
´10
.
ex sin3 x
.
(x − 2)9
´
13. Find the equation of the line tangent to y = ln x2 + 1 at the point (1, ln 2). Find the
equation of the normal line at the point (1, ln 2).
µ
14. Find the equations of the lines tangent to y =
15. Find a formula for f (n) if f (x) = ln x.
¶
ln x
1
at the points (1, 0) and e,
.
x
e
In exercises 1-10 find the derivative.
³
√ ´
1. y = Arctan Arcsin x
2. y = Arcsec (2x )
³
´
3. y = Arcsin x2 + 2x + 3
√
4. y = Arctan x5 − 2x + 5
³
5. y = Arcsin x3 + xπ + π x
³
6. y = x2 Arctan x2
7. y =
´
´
Arcsin (x2 )
x2 + 1
µ
Arcsin x
8. y =
1 + Arccos x
³
9. y = Arcsin 1 − x2
³
¶2
´
´
10. y = Arccsc2 x2 + 3x − 1
In exercises 11-22 find the angle.
µ ¶
1
11. Arcsin
2
√
14. Arcsec( 2)
³ √ ´
17. Arccsc − 2
20. Arccot (1)
Ã
1
12. Arcsin − √
2
Ã
2
15. Arcsec − √
3
Ã
2
18. Arccsc √
3
Ã√ !
!
3
2
13. Arcsin
!
16. Arcsec(2)
!
³ √ ´
21. Arccot − 3
19. Arccsc (−2)
Ã
1
22. Arccot √
3
!
1. Gas in pumped into a spherical balloon at the rate of 4π m3 /sec. What is the rate of change
of the radius of the balloon when the radius of the balloon is 4 m? What is the rate of change
of the surface area of the balloon when the radius of the balloon is 4 m?
2. The base of an isosceles triangle is a constant length of 3 m and its area is changing at a rate
of 3 m2 /sec. At what rate is the altitude changing when the area is 6 m2 ? What is the rate
of change of the perimeter when the area is 6 m2 ?
3. The dimensions of a right circular cylinder are changing with time. When the radius is 1 cm
and the height is 3 cm, the radius is getting larger at a rate of 3 cm/sec and the height is
getting smaller at a rate of 2 cm/sec. At what rate is the volume of the cylinder changing?
At what rate is the surface area (including the top and bottom) changing?
√
4. A particle is moving along the curve y = x. As the particle passes through the point (4, 2),
its x–coordinate is increasing at a rate of 3 cm/sec. How fast is the distance from the particle
to the origin changing at this instant?
5. A kite 100 ft above the ground moves horizontally at a speed of 8 ft/sec. At what rate is the
angle between the string and the ground decreasing when 200 ft of string have been let out?
6. A plane flying with constant speed of 300 km/hr passes over a ground radar station at an
altitude of 1 km. How fast is the distance from the plane to the radar station changing after
1 hour?
7. A paper cup has the shape of a cone with height 10 cm and radius 3 cm (at the top). If water
is poured into the cup at a rate of 2 cm3 /sec, how fast is the water level rising when the water
is 5 cm deep?
8. A balloon is rising at a constant speed of 5 ft/sec. A boy is cycling along a straight road at
a speed of 15 ft/sec. When he passes under the balloon it is 45 feet above him. How fast is
the distance between the boy and balloon increasing 3 seconds later?
MATHEXCEL 1611 WORKSHEET RELATED RATES
1. A television camera is positioned 4000 ft from the base of a rocket launching pad. The angle
of elevation of the camera has to change at the correct rate in order to keep the rocket in
sight. Also, the mechanism for focusing the camera has to take into account the increasing
distance from the camera to the rising rocket. Let’s assume the rocket rises vertically and its
speed is 600 ft/s when it has risen 3000 ft.
(a) How fast is the distance from the television camera to the rocket changing at that moment?
(b) If the television camera is always kept aimed at the rocket, how fast is the camera’s angle
of elevation changing at that same moment?
2. A lighthouse is located on a small island 3 km away from the nearest point P on a straight
shoreline and its light makes four revolutions per minute. How fast is the beam of light moving
along the shoreline when it is 1 km from P ?
3. A plane flying with a constant speed of 300 km/h passes over a ground radar station at an
altitude of 1 km and climbs at an angle of 30◦ . At what rate is the distance from the plane
to the radar station increasing a minute later?
4. The volume of a cube is increasing at a rate of 10 cm2 /min. How fast is the surface area
increasing when the length of an edge is 30 cm?
5. A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first
base with a speed of 24 ft/s.
(a) At what rate is his distance from second base decreasing when he is halfway to first base?
(b) At what rate is his distance from third base increasing at the same moment?
1. The radius of a circular disk is given as 24 cm with a maximum error in measurement of 0.2
cm. Use differentials to estimate the maximum error in the calculated area of the disk. What
is the relative error?
2. Use differentials to estimate the amount of paint needed to apply a coat of paint 0.05 cm thick
to a hemispherical dome with diameter 50 m.
√
3. Find the linear approximation to f (x) = 25 − x2 near 3.
√
√
4. Find the linearization of f (x) = 3 1 + 3x at a = 0. Use it to approximate the value of 3 1.03.
5. Evaluate dy if y = x3 − 2x2 + 1, x = 2, and dx = 0.2.
6. A window has the shape of a square surmounted by a semicircle. The base of of the window
is measured as having width 60 cm with a possible error in measurement of 0.1 cm. Use
differentials to estimate the maximum error possible in computing the area of the window.
7. Use differentials to estimate ln 1.07.
8. Use differentials to estimate (1.97)6 .
2
9. Find the linear approximation for f (x) = ex at a = 1.
10. Find the linear approximation for f (x) = tan x at a = 0.
11. y = sin x, x = π/3, and dx = 0.2. Evaluate dy.
12. Find the differential of the function y = (1 + 2x)−4 .
13. Find the differential of the function y = cos(πx).
14. Find the linearization of the function f (x) = e−x sin x at a = 0.
MATHEXCEL 1611 WORKSHEET SECTIONS 4.1-2
In exercises 1–7 find the absolute extrema, both x and y coordinates.
3
1. f (x) = ex −x ,
−1 ≤ x ≤ 0
√
2. f (x) = x x − x2
Find the domain first.
3. f (x) =
cos x
,
2 + sin x
4. f (x) = x3 − 8x + 1,
0 ≤ x ≤ 2π
−3 ≤ x ≤ 3
5. f (x) = x2 e−x ,
0≤x≤3
√
6. f (x) = x − 2 sin x,
0≤x≤π
7. f (x) =
x2
x
,
+x+1
−2 ≤ x ≤ 0
8. Show that the equation x5 + 10x + 3 = 0 has exactly one real root.
µ
9. Show that the equation 3x − 2 + cos
¶
πx
= 0 has exactly one real root.
2
10. Show that the equation x101 + x51 + x − 1 = 0 has exactly one real root.
11. Suppose that f is continuous on [0, 4], f (0) = 1, and 2 ≤ f 0 (x) ≤ 5 for all x in (0, 4). Show
that 9 ≤ f (4) ≤ 21.
12. Does there exist a function f such that f (0) = −1, f (2) = 4 and f 0 (x) ≤ 2 for all x?
13. A number a is called a fixed point of a function f if f (a) = a. Prove that if f 0 (x) 6= 1 for
all real numbers x, then f has at most one fixed point.
MATHEXCEL 1611 WORKSHEET SECTION 4.3-4
In the following exercises:
• Find the critical points, both x and y values.
• Find where f (x) is increasing/decreasing.
• Find the relative maxima and minima.
• Find where the second derivative is zero or undefined.
• Find where f (x) is concave up/down.
• Find the inflection points, both x and y values.
• Use the above information to sketch the function.
1. f (x) = x3 + 2x2 − 4x + 1
2. f (x) = (x + 6)2 (x − 2)2
3. f (x) = (x + 6)2 (x − 2)3
4. f (x) = sin x + cos x, on 0 ≤ x ≤ 2π
5. f (x) = sin2 x, on 0 ≤ x ≤ 2π
6. f (x) = e−x cos x, on 0 ≤ x ≤ 2π
In the following, also find the asymptotes.
x2 + 1
x2 − 1
x
8. f (x) = 2
x +1
7. f (x) =
9. f (x) =
x2 − 4
x+4
x2 − 4
10. f (x) =
x+1
11. f (x) =
x2 − 4
x+2
12. f (x) =
x2 − 4
x
13. f (x) = x3 −
24
x
√
1. You are given the semicircle y = 25 − x2 , centered at the origin, and lying above the x–axis.
Find the area of the largest rectangle which can be inscribed under this semicircle, with one
side lying on the x–axis.
2. Find the dimensions of the rectangle of maximum area that can be inscribed in the region
bounded between the x–axis and the parabola y = −x2 +4. Note that one side of the rectangle
must lie on the x–axis.
3. Find the point on the line 6x + y = 9 that is closest to the point (−3, 1).
4. You have a river that is 5 miles wide and 10 miles long. You are going to run along the bank
of the river to a point P and then swim diagonally across the river to get to the other bank
at the end of the river. If you can swim at a rate of 3 mph and run at a rate of 5 mph, where
should the point P be in order to minimize the time of this trip?
5. If a widget manufacturer produces 80 widgets a day, he can sell them all at $65 each. For
each $2 increase in unit price, he sells 1 less widget. Each widget costs $10 to produce. What
price optimizes his total profit?
6. Dirty Sam is lost in the desert. At one point, a search party is 5 mi directly west of Sam.
The search party is proceeding East at 4 mi/hr, while Sam is crawling north at 1 mi/hr. How
close does the search party come to finding Sam?
7. A 3 ft by 4 ft piece of cardboard has a square cut out of each corner, and the edges are then
folded up to make an open box. Find the dimensions of the box with maximum volume.
8. Several inmates plan to break out of prison. They are housed in a 100 ft high building without
windows. The building is surrounded by a 15 ft tall fence, 10 ft from the building. Help from
the outside is to provide a ladder that is to pass over the fence and lean against the building.
The inmates will lower themselves by a rope from the top of the building to the top of the
ladder. What is the shortest ladder that will suffice?
Evaluate the following limits.
1. lim
sin 3x
sin 2x
2. lim
sin(x) − x
x3
x→0
x→0
cos(x) − 1 − (x2 /2)
x→0
x4
3. lim
ex − 1 − x
x→0
x2
4. lim
ln(1 + x) − x + (x2 /2)
x→0
x3
5. lim
6. lim
x→0
1 + sin x − cos x
1 − sin x − cos x
7. x→∞
lim
ln (ln x)
ln x
1
8. lim x x−1
x→1
µ
9. lim
x→π/2
µ
a
1+
x
10. x→∞
lim
¶
π
− x tan x
2
¶bx
11. lim (ex + x)1/x
x→∞
µ
12. x→∞
lim
2x − 3
2x + 5
¶2x+1
µ
1
1
13. lim
−
x→1 ln x
x−1
³
14. lim xe1/x − x
¶
´
x→∞
³
15. lim csc2 x − x−2
x→0
eax − ebx
x→0
x
16. lim
´
In exercises 1–6 evaluate the integrals.
Z
x1/2 + x−1/2
dx
x
Z
1
dx
2 + 2x2
1.
2.
Z
(1 + x)2 dx
3.
4.
Z ³
1 + x2
´2
dx
Z
5.
(x + 1)(2x − 3) dx
Z
6.
ex + sec x tan x dx
µ ¶
1
4
7. Solve the initial value problem f (x) = √
, where f
= 1.
2
2
1−x
√
8. Solve the initial value problem f 0 (x) = x + x, where f (1) = 2.
√
9. Solve the initial value problem f 00 (x) = x + x, where f (1) = 1, f 0 (1) = 2.
0
10. Solve the initial value problem f 00 (x) = 3ex + 5 sin x, where f (0) = 1, f 0 (0) = 2.
11. Solve the initial value problem f 000 (x) = sin x, where f (0) = 1, f 0 (0) = 1, and f 00 (0) = 1.
12. Find the position function given that v(t) = 2t1/2 where s(1) = 4.
13. Find the position function given that a(t) = cos t + sin t where s(0) = 0, and v(0) = 5.
14. A car is traveling at 50 mi/hr when the brakes are fully applied producing a constant deceleration of 40 ft/s2 . What is the distance covered before the car comes to a stop?
In exercises 1-4 use finite approximations to estimate the area under the graph of the function
using
a. a lower sum with two rectangles of equal width
b. a lower sum with four rectangles of equal width
c. an upper sum with two rectangles of equal width
d. an upper sum with four rectangles of equal width
1. f (x) = x3 between x = 0 and x = 1
2. f (x) = 4 − x2 between x = −2 and x = 2
3. f (x) = sin x between x = 0 and x = π
4. f (x) = ln x between x = 1 and x = 5
In exercises 5-8 use rectangles whose height is given by the value of the function at the midpoint
of the rectangle’s base (use the midpoint rule) to estimate the area under the graph of the function,
using first two and then four rectangles.
5. f (x) = 4x between x = 1 and x = 3
6. f (x) = 16 − x2 between x = −4 and x = 4
7. f (x) = ex between x = 0 and x = 8
8. f (x) = x2 + x between x = 0 and x = 2
In exercises 1-9 evaluate the expressions.
1.
8
X
i(i + 2)
i=3
2.
100
X
4
i=1
3.
5
X
(2 − 5i)
i=1
4.
4
X
(3 + 2i)2
i=1
5.
3
X
i(i + 1)(i + 2)
i=1
6.
4
X
n2
n=−2
7.
3
X
(k + 1)π
k=−1
8.
4
X
k!
k=1
9.
7
X
2k
k2
(Be careful!)
n=1
10. Express 1 − 2 + 4 − 8 + 16 − 32 in sigma notation.
11. Express
1 3 5
7
− + −
in sigma notation.
2 4 8 16
12. Express
4 8 16 64
+ +
+
in sigma notation.
3 9 27 81
For the functions in exercises 13-15 find a formula for the upper sum obtained by dividing the
interval [a, b] into n equal subintervals. Then take a limit for these sums as n → ∞ to calculate the
area under curve over [a, b].
13. f (x) = 2x over the interval [0, 3]
14. f (x) = 3x2 over the interval [0, 1]
15. f (x) = x2 + x over the interval [1, 3]
1. Evaluate
2. Evaluate
3. Evaluate
Z 3
−1
Z 3
0
Z 1
1
(2 − x) dx by interpreting it as an area.
|3x − 5| dx by interpreting it as an area.
x2 cos x dx.
n
X
exi
∆xi on [0, 1] as a definite integral.
kP k→0
i=1 1 + xi
4. Express lim
n
X
5. Express lim
kP k→0
·
sin xi ∆xi on 0,
i=1
¸
π
as a definite integral.
2
6. Use the limit form of the Riemann sum to evaluate the integral
7. Use the limit form of the Riemann sum to evaluate the integral
8. If
Z 1
0
f (x) dx = 2,
Z 4
0
f (x) dx = −6, and
Z 4
3
11. Estimate
12. Estimate
13. Estimate
14. Estimate
Z 2√
0
Z 2
0
Z b
a
f (x) dx ≤ M (b − a).
x3 + 1 dx.
xe−x dx.
Z π/3
π/4
cos x dx.
Z 3π/4
π/4
sin2 x dx.
15. Find the average value of f (x) = −
1
Z 5³
f (x) dx = 1, find
Recall, if m ≤ f (x) ≤ M for a ≤ x ≤ b then
m(b − a) ≤
Z 3³
x2
on [0, 3]
2
16. Find the average value of f (x) = 3x2 − 3 on [0, 1]
17. Find the average value of f (x) = x2 − x on [−2, 1]
0
Z 3
1
2 + 3x − x2
1 + 2x3
´
f (x) dx.
´
dx.
dx.
1.
5.
9.
10.
d Z cos x 1
dt
dx 0
t3 + 1
Z 3
2
(x + 1)3 dx
Z π
−π
6.
Z 2 2
x + x3
x
1
(
f (x) dx
Z ln 6
ln 3
d Z1 1
dt
dx x2 t3 + 1
2.
where f (x) =
x
8e dx
Z π/3
11.
dx
7.
Z 1
0
x
sin x
2
csc x dx
π/6
3.
d Z cos x 1
dt
dx x2 t3 + 1
√
(3 + x x) dx
8.
4.
Z 2
1
Z 2
1
(x + 1)(x + 2) dx
x
3 + √ dx
x
if − π ≤ x ≤ 0
if 0 < x ≤ π
12.
Z 1/2
0
√
1
dx
1 − x2
13.
Z e3
3
e2
x
dx
14. Find the interval on which the curve
y=
Z x
0
1
dt
1 + t + t2
is concave up.
Z
15.
sin 2x
dx
sin x
Z 2 ¯
¯
¯
¯
18.
¯x − x2 ¯ dx
−1
21.
Z −2 4
x −1
−5
x2 + 1
Z 2 6
x − x2
16.
dx
x4
1
19.
22.
Z 2³
0
Z −1
1
2
dx
17.
Z 2³
0
´
x − |x − 1| dx
(x − 1)(3x + 2) dx
´2
x3 − 1
20.
dx
Z 0 Ã√
23.
4
Z 2π
0
1
x+ √
x
!2
dx
| sin x| dx
Z 8¯
¯
¯ 2
¯
24.
¯x − 6x + 8¯ dx
0
25. Find the area of the region bounded by x = 0 and the function x = 2y − y 2 . Hint: The
function touches the y–axis at y = 0 and y = 2. Integrate the function with respect to y.
√
26. Find the area of the region bounded by x = 0, y = 1 and y = x. Do this two ways: first, by
integrating with x as the variable and secondly, by integrating with y as the variable.
27. The velocity function is given by v(t) = t2 −2t−8 on the interval 1 ≤ t ≤ 6. Find the distance
traveled.
28. The acceleration function is given by a(t) = 2t + 3. The initial velocity is v(0) = −4. Find
the distance traveled when 0 ≤ t ≤ 3.
In exercises 1–11 evaluate the integrals.
Z
1.
x
dx
1 + x4
Z
√
2.
x2
dx
1−x
Z
tan2 x sec2 x dx
3.
4.
5.
6.
7.
Z 4
√
0
x
dx
1 + 2x
Z 1/2
sin−1 x
√
0
Z π/2 2
x sin x
1 + x6
−π/2
Z π/3
−π/3
dx
1 − x2
dx
sin5 x dx
Z
cos x cos(sin x) dx
8.
Z
q
sin(x − 1) cos(x − 1) 1 + sin2 (x − 1) dx
9.
µ
Z
¶
µ
3x + 1
3x + 1
10. sin
cos
2
2
√
Z
cos x
√
√ dx
11.
x sin2 x
3
¶
dx
12. The substitution u = tan x gives us
Z
Z
u2
1
2
sec x tan x dx = u du =
+ C = tan2 x + C.
2
2
The substitution u = sec x gives us
Z
Z
2
sec x tan x dx =
Z
sec x(sec x tan x) dx =
Is something wrong here? Explain.
u du =
u2
1
+ C = sec2 x + C.
2
2
1. Evaluate
2. Evaluate
3. Evaluate
4. Evaluate
5. Evaluate
Z 0
√
−1
Z 4
2
x3
dx.
x4 + 9
dx
.
x ln x
Z pi/3
4 sin θ
dθ.
1 − 4 cos θ
0
Z −1/2
−1
Z π/6
0
µ
−2
t
sin
2
1
1+
t
¶
dt.
(1 − sin 2t)3/2 cos 2t dt.
In exercises 6–11 find the area bounded by the curves using
6. y = x, y = x2 .
9. y = x1/3 , x = y 2 .
R
. . . dx and then using
7. y = x2 , y = x4 .
√
10. y = x, y = 0, and y = x − 2.
R
. . . dy.
8. y = |x|, y = x2 − 2.
11. y = 2x, y = x2 − 4.
12. Find the area bounded by x = y 2 and x = y + 2 in the first quadrant.
13. Find the area bounded by y = sin x and y = cos x from x = π/4 to x = 5π/4.
14. Find the area bounded by the curves x = y 3 − y 2 and x = 2y.
15. Find the area bounded by the curves y = tan2 x and y = − tan2 x from x = −π/4 to x = π/4.
From the geometry of the problem, simplify the statement of the integral as much as possible.
√
16. Find the area bounded by the curves x = 3 sin y cos y and x = 0 from y = 0 to y = π/2.
17. Find the area bounded by the curves y = x − 1 and y 2 = 2x + 6 by integrating with respect
to x.
18. Find the area bounded by the curves y = sin x and y = ex from x = 0 to x = π/2. Which
curve is on top? Why? (Your answer must include information about increasing or decreasing
functions.)
19. Find the area bounded by the curves y = 2 − x2 and y = e−x from x = 0 to x = 1. Which
curve is on top? Why? (Your answer must include information about increasing or decreasing
functions.)
20. Find the area of the region bounded by the parabola y = x2 , the tangent line to this parabola
at (1, 1) and the x–axis.
21. Find the number b such that the line y = b divides the region bounded by the curves y = x2
and y = 4 into two regions with equal area.
MATHEXCEL 1611 WORKSHEET FINAL REVIEW
PRACTICE FINAL EXAM 1
x2 − 2x − 3
.
x→3
x−3
1. Find the derivative of y = sin(x) cos(x).
2. Evaluate lim
Z
sin x + csc2 x dx
3. Evaluate
4. Evaluate
³ ´
d Z x3
sin t2 dt
dx 1
f (2 + h) − f (2)
.
h→0
h
5. Find the derivative of x2 +y 2 = xy+1 at (1, 1).
6. f (x) = 4x−1 . Evaluate lim
7. Find the x–coordinate of the point on the line y = x − 6 that is closest to (1, 1).
8. A ladder 10 ft long is leaning against a wall, with the foot of the ladder 8 ft from the wall.
If the foot of the ladder is being pulled away from the wall at 3 feet per second, how fast, in
feet per second, is the top of the ladder sliding down the wall?
10. Find the derivative of y = sin3 x.
9. Let y = x4 − 4x3 . Where is y decreasing?
0
11. Solve f (x) = 1 + x
−2
for f (x) when f (1) = 1.
12. Evaluate
Z 1
−2
4x3 + 6x2 − 2 dx.
13. Find the general antiderivative of 3x−5/2 .
14. Find the horizontal and vertical asymptotes for y =
1 + 4x2 − 2x3
.
x4 − 3x3
15. Find the absolute max/min of the function f (x) = sin x on the interval [−π/2, 3π/4].
16. f (x) = −x3 + 6x2 and f 0 (x) = −3x2 + 12x. Where is f (x) concave up?
x2 + x
17. Evaluate lim−
.
x→1
x−1
Z
ex − 1 − x
.
x→0
x2
20. Evaluate lim
22. Evaluate
24. Evaluate
Z 1/2
0
√
1
dx.
1 − x2
Z 2 2
x −1
dx.
x
dx.
1 − x2
19. y = xex . Find y 00 .
³
´
21. Find the derivative of y = csc−1 x3 .
23. Find the derivative of y =
Z
x2
√
x+3−2
27. Evaluate lim
x→1
x−1
1
√
18. Evaluate
25. Evaluate
cos(ln x)
dx.
x
ex
.
x
26. Evaluate lim
28. Sketch the graph with the following properties:
• Increasing 0 < x < 1, x > 1.
• Concave down x < −1,
x > 1.
Decreasing x < −1, −1 < x < 0.
Concave up −1 < x < 1.
• Vertical Asymptotes x = −1, x = 1. Horizontal Asymptote y = 1.
• (0, 0) is a relative minimum. (−2, 0) and (2, 0) are points on the graph.
x→0
1
(1+x)2
x
−1
.
x2 + 2x − 3
.
x→−3 x2 + x − 6
1. Find the derivative of y = x3 tan x.
2. Evaluate lim
Z
d Z1 1
4. Evaluate
dt
dx x2 t2 + 1
2
3. Evaluate
cos x − sec x dx
5. Find the derivative of xy+y 2 = x. Solve for y 0 .
6. f (x) =
√
f (9 + h) − f (9)
.
h→0
h
x. Evaluate lim
7. A rancher has 240 feet of fencing to enclose two adjacent rectangular corrals. Find the
maximum area.
y
y
y
x
x
8. An airplane flying at an altitude of 6 miles passes directly over a radar antenna. When the
airplane is 10 miles away from the antenna, the radar detects that the distance is changing at
a rate of 240 miles per hour. What is the horizontal speed of the airplane?
9. Let y = x3 − 3x. Where is y increasing?
10. Find the derivative of y = sec3 x.
11. Solve f 0 (x) = x3 + x for f (x) when f (1) = 1.
12. Evaluate
Z 1
−1
4x3 + 4x + 1 dx.
13. Find the general antiderivative of x−3/2 + x2/3 .
1 + 7x2 + 3x3
.
x3 − x
15. Find the absolute maximum and minimum of the function f (x) = cos x on the interval
[−π/2, 3π/4].
16. f (x) = x4 − 12x3 and f 0 (x) = 4x3 − 36x2 . Where is f (x) concave up?
Z
1
17. Evaluate lim
.
x→−5 (5 + x)5
sin x − x
.
x→0
x3
22. Evaluate lim
Z
25. Evaluate
x
dx.
1 + x4
ex sin (ex ) dx.
18. Evaluate
ln x
.
x
x2 + 2
. Find y 00 .
x
19. y =
³
Z 1
23. Evaluate
0
26. Evaluate lim
x→3
1
dx.
x2 + 1
1
5−x
− 12
.
3−x
Z
x4 + x2
dx.
x
√
x+4−2
27. Evaluate lim
.
x→0
x
24. Evaluate
• Increasing −1 < x < 0,
x>1
Decreasing x < −1,
0 < x < 1.
• Relative maximum at (0, 0).
´
21. Find the derivative of y = sin−1 x2 .
0 < x < 1,
x > 1.
• (0, 0) relative maximum. (−2, 0) and (2, 0) are points on the graph.
√
1. Find the derivative of y = x 1 − x2 .
x2 + x − 12
.
x→−4 x2 + 7x + 12
2. Evaluate lim
Z
sin x − csc2 x dx
3. Evaluate
d Z 3x 1
5. Evaluate
dt
dx 2x ln t
4. Find the derivative of x2 y 2 + x3 + y = 8 at (2, 0).
f (0 + h) − f (0)
.
h→0
h
6. f (x) = e2x . Evaluate lim
7. Find the x value of the point on the curve y = 3x + 2 closest to the point (1, 0).
8. An icicle in the shape of a cone is growing in volume at the rate of 1 cm3 /min. The height
always equals twice the radius of the base. When the height equals 10 cm, how fast is the
height increasing? (Hint: V = 31 πr2 h.)
9. Let y = xex . Where is y increasing?
10. Find the derivative of y = cot−2 (x).
11. Solve f 0 (x) = 3x2 + ex for f (x) when f (0) = 2.
12. Evaluate
Z 2
1
3x2 − 2x dx.
13. Find the general antiderivative of 3x1/2 + x−1/2 .
1 + 7x2 + 3x3
.
x4 − x
15. Find the absolute maximum and minimum of the function f (x) = x4 − 8x2 on the interval
−1 ≤ x ≤ 3.
16. f (x) = x4 − 6x2 and f 0 (x) = 4x3 − 12x. Where is f (x) concave up?
x
.
x→−4 (4 + x)6
17. Evaluate lim
Z
√
18. Evaluate
³
´
20. Find the derivative of y = Arctan x2 .
ln x − x + 1
22. Evaluate lim
.
x→1
(x − 1)2
3x2
25. Evaluate
dx.
1 − x6
√
3 − 12 − x
27. Evaluate lim
x→3
3−x
19. y =
x3 + 3x
. Find y 00 .
x
23. Evaluate
Z 1/2
√
0
Z
√
2x3
dx.
x4 + 9
26. Evaluate lim
1
dx.
1 − x2
ln2 x
.
x
Z
24. Evaluate
1
3−x
x→1
1
− x+1
.
x−1
• Increasing −1 < x < 0,
0 < x < 1,
−1 < x < 0,
x>1
x > 1,
Decreasing x < −1.
Concave up 0 < x < 1.
• (0, 0) is an inflection point. (−2, 0) and (2, 0) are points on the graph.
9x4 + 5x2
dx.
x1/2

MATHEXCEL 1611 WORKSHEET FIRST REVIEW

Transcription

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