Solution

Advice for studying for exams
• First and most important thing to do: past midterm exams. Each third of the
semester is worth 50 points, but M1 material is much easier than M3, so don’t
neglect the easy topics. Do at least past 3 semesters of each of the three midterm,
but past 5 would be even better.
• Flashcards.
– General form of an exponential function
– Logarithm identitties
– Quotient rule
– Trig derivatives
– Trig identities, double angle etc
– IVT, Rolle, MVT
– Newton recurrence
– FTC1, FTC2
– Volume formulas
– Hyperbolic functions & their derivatives
– Integration by parts formula
– etc.
d
Cut out small paper cards, on one side write e.g. “ dx
cot x” and on the other side
2
write “− csc x”. For this course, you probably won’t need more than 30 of them.
Repeatedly go through them until you score at least 95%. After that, review once
a day. Fun & really useful for memorizing stuff.
• Feel free to e-mail me or any other TA with questions the night before the final.
In particular, George Shakan *loves* to be e-mailed & called at midnight.
Midterms 1 and 2 Review + Sections 3.11 & 7.1
1.
[1.2 −√1.3] Use interval notation to express the domain of the given functions.
x2 − 1
ln(x − 3)
(a) f (x) =
(b) g(x) =
(c) h(x) = arcsin(3x)
2
3x + 2
2x2 − 8
Solution: (a) Need to check whether we divide by zero, but 3x2 + 2 is always positive, so
only need to worry about the top. Need x2 − 1 ≥ 0 so domain is (−∞, −1] ∪ [1, ∞).
(b) - Need x − 3 > 0 so x > 3. For the bottom, need 2x2 − 8 6= 0 so x 6= ±2 but these
were already excluded. So domain is (3, ∞).
(c) - Domain of arcsin is [−1, 1] so domain is [−1/3, 1/3].
2.
[1.3] Sketch the graphs
√ √ of the√given functions.
√
√ √
(a) x, x + 2, 2x
(b) x, x − 2, x − 2, − x, 2 x, −x
(d) sin(2x), cos(x + π)
(c) ex , e2x , e · ex , ex+1
Use wolframalpha / examples in book to check your solutions.
3.
[1.5] The number of ducks in Urbana increases exponentially as a function of time.
Given D(2) = 10 and D(5) = 90, determine a formula for D(t).
Solution: General formula is D(t) = Cat (or D(t) = Ceat if you prefer). Have two
equations for two unknowns: D(2) = 10 = Ca2 and D(5) = 90 = Ca5 . Many ways to
solve these. One way is to divide second equation by first, so 9 = a3 and hence a = 32/3
2t/3 .
and so C = 10/34/3 , so D(t) = 310
4/3 3
[1.6] Find a formula for f −1 (x) given the function f (x).
x−3
6ex
1
(b) f (x) =
(c) f (x) =
(a) f (x) =
x+2
2x + 5
2 + 9ex
4.
1
Solution: (a) Solve x = y+2
, get f −1 (x) =
(b) f −1 (x) = −5x−3
2x−1 .
−1
(c) f (x) = − log 6−9x
2x
1−2x
x .
5.
[1.6] Solve the following equation for x and simplify your answer.
(a) ln x = 4 ln 2 − 2 ln 3
(b) 2e2x = 5ex−1
(c) e3−2 ln x = 4e3
Solution:
(a) Combine logs on right hand side, then exponentiate both sides to get x = 16/9.
(b) Take logs of both sides, get x = −1 − log(2) + log(5).
(c) e3 /x2 = 4e3 so x = ± 12 , but need x > 0 since have ln x. So x = 1/2.
6.
[2.2 − 2.6]Evaluate the following limits without the use of derivatives. Show
sufficient justification for each answer. An answer of ’does not exist’ is not sufficient. For
infinite limits you must state if it is ∞ or −∞.
1
2x2 − 10x + 8
x2 + 2x
(a) lim √
(c) lim 2
(b) lim 2
x→−3
x→−2 x + 4x + 4
x→4 x − x − 12
3x2 − 2 − x2
x2 − 3x
2 − 3x + x2
ln 9 + x2
(d) lim
(e)
lim
(f
)
lim
x→∞ 2 − 3x − 4x2
x→−∞
1 − x3
x→−3+ ln 9 − x2
Done in class.
7.
Find all horizontal & vertical asymptotes of f (x) =
16+5e2x
3e2x −8
Done in class.
8.
[3.1 − 3.4] Differentiate the following functions.
(a) f (x) = 2x3 − e3x (b) g(x) = ln(5x) + tan2 x + cos x (c) h(x) = 2 csc(2x) sec(3x)
4
2
cos(x2 + 2)
x +2
(d) F (x) =
(e) G(x) =
(f ) H(x) = ln(cot(cos x))
4x2 − sin x
x3 − 1
Solution: (I just plugged these into wolframalpha..)
(a) 6x2 − 3e3x
(b) x1 + 2 tan(x) sec2 (x) − sin(x)
(c) 6 tan(3x) csc(2x) sec(3x) − 4 cot(2x) csc(2x) sec(3x)
(d) −(2x sin(x2 + 2))/(4x2 − sin(x)) − ((8x − cos(x)) cos(x2 + 2))/(4x2 − sin(x))2
(e) −(4x(x2 + 2)3 (x3 + 6x + 2))/(x3 − 1)5
(f) sin(x) csc(cos(x)) sec(cos(x))
9.
[3.5] Find an equation of the tangent to the circle x2 + y 2 = 25 at the point
3, 4.
Done in class.
10.
[3.8] In this problem, both f (x) and g(x) are differentiable.
df
(x) = 4x. If f (2) = 3, find a formula for f (x).
(a) Suppose f (x) satisfies dx
dg
(b) Suppose g(x) satisfies dx (x) = −2g(x). If g(3) = 5, find a formula for g(x).
Solution:
(a) f (x) = 2x2 + c, and 3 = 2 · 4 + c gives c = −5 so f (x) = 2x2 − 5.
(b) g(x) = ce−2x and 5 = ce−6 gives c = 5e6 so g(x) = 5e6−2x .
11.
[4.1] Determine the absolute minimum y-value on the graph y = 2x2 −ln x+6.
Solution:
Domain is x > 0. Differentiate, set equal to zero: 0 = 4x − x1 so x = ± 12 , only x = 1/2
is in the domain. If x < 1/2 the derivative is negative, if x > 1/2 derivative is positive
so x = 1/2 is global minimum.
12.
[4.3] A polynomial p(x) has the following second derivative:
p00 (x) = 6(x − 7)10 (x + 1)3 (18 − 2x2 ).
Find the intervals of concavity for p(x). State each x-value at which the graph of p(x)
has an inflection point.
Soltuion: Concave up on (−∞, −3) ∪ (−1, 3). Concave down on (−3, −1) ∪ (3, ∞).
Inflection pt at −3, −1, 3.
13.
[3.9] The top of a 20-foot ladder slides down a vertical wall at a rate of 4 ft/s. At
the moment when the bottom of the ladder is 16 ft from the wall, how fast is the bottom
of the ladder moving away from the wall?
Solution: Let y be the vertical distance, x the horizontal distance, so dy/dt = −4ft/s.
dy
Have x2 + y 2 = 202 , differentiate this to get 2x dx
dt + 2y dt = 0. When x = 16ft have
y = 12ft so 2 · 16 · dx
dt = 2 · 12 · 4 giving dx/dt = 3ft/s.
14.
[4.4] Evaluate the following limits.
2x3 − 5x2 + 3
3x + 4 − 4ex
(a) lim
(b)
lim
x→1 3x4 + x2 − 4
x→0 ln(x2 + e) − 1 + 3x
x2
x→0 e−4x + 4x − cos x
(c) lim
Done in class (?). Use L’Hopital’s rule, (a) −2/7, (b) −1/3, (c) 2/17.
15.
[?] Let f (x) = 5x2 − 4x. Use the definition of a derivative as a limit to prove
that
= 10x − 4. Show each step in your calculation and be sure to use proper
terminology in each step of your proof.
f 0 (x)
Solution: Check midterm 1 solutions (question 1) on the course website.
16.
[3.11] Write down the definitions of the following functions:
sinh(x) =
(b) cosh(x) =
(c) tanh(x) =
Then, find the derivatives of the following functions:
(a) f (x) = 3 sinh(x) − 4 cosh(x)
(b) g(x) = cosh(x) sinh(cosh(x))
sinh(x)
(c) h(x) = ln(cosh x)e
Solution:
x
−x
x
−x
sinh(x)
, cosh(x) = e +e
, tanh(x) = cosh(x)
sinh(x) = e −e
.
2
2
(a) 3 cosh(x) − 4 sinh(x)
(b) sinh(x)(cosh(x) cosh(cosh(x)) + sinh(cosh(x)))
(c) esinh(x) (tanh(x) + cosh(x) log(cosh(x)))
17.
[7.1] (Integration by Parts) Fill in the blank: If u, v are differentiable functions,
Z
Z
udv = uv − vdu
Then,
to find the following
indefinite integrals:
Z use integration by parts
Z
Z
2 2x
x
(a)
7x sin x dx
(b)
x e dx
(c)
e sin x dx (assigned in class)
Solution:
R
R
(a) u = 7x, v = sin xdx gives 7x sin x dx = −7x cos x + 7 cos xdx = −7x cos x +
7 sin x + C
(b) uR = x2 , integrate by parts twice,
get 14 e2x (2x2 − 2x + 1) + C.
R
R x
x sin x dx = ex sin x −
x cos x dx = ex sin x − ex cos x −
(c)
e
e
e sin x dx, so
R x
e sin x dx = 12 ex (sin x − cos x) + C.