STMATH 124: FINAL EXAM REVIEW PROBLEMS (1) Evaluate each

STMATH 124: FINAL EXAM REVIEW PROBLEMS
(1) Evaluate each limit.
x−4
(a) lim 2
x→4 x − x − 12
2x3
(c) lim 2
x→−∞ x + 1
3x − 2
x→∞ 9x − 7
2x
(d) lim−
x→3 9 − x2
(b) lim
Use the graph of g below to evaluate each limit.
(e) lim− g(x)
x→5
(g) lim+
x→6
x−5
g(x)
g(4 + h) − g(4)
h→0
h
(h) lim g(x)
(f) lim
x→6
(2) List all vertical and horizontal asymptotes of y =
x2 − x
.
x2 − 3x + 2

 x2 − 8 if x ≤ a
(3) Let f =
Find all values of a so that f is continuous for all x.
x 2 − a2

if x > a
x−a
(4) Find the equation of the tangent line to the curve y =
(5) Find
√
9 − 4x at x = −4.
dy
for each of the following.
dx
(a) y = (2x3 + 5x2 − 6x − 4)5
r
x
(c) y =
3x + 1
(b) y = tan3 (5x)
(d) y = e2x sin x
(e) 4x2 y + x2 − 6y 4 = 5
(f) y = arctan(x3 )
1
(g) y = ln
(5x + 2)4
√
x2 + 5
√
(h) y = (tan x)
x
(6) Use the graph of f (x), shown below, to determine if each of the following is positive, negative,
or zero.
(i) f (a)
(iv) f 0 (a)
(vii) f 00 (a)
(ii) f (b)
(v) f 0 (b)
(viii) f 00 (b)
(iii) f (c)
(vi) f 0 (c)
(ix) f 00 (c)
(7) Use the first and second derivatives to determine all the extrema and inflection points for the
curve y = x5 − 30x3 + 6.
(8) Discuss the concavity of the function y = x3 − 6x2 + 9x.
(9) Analytically determine the absolute minimum and the absolute maximum of f (x) = x3 − 2x2
on [−1, 1].
(10) A power company needs to lay a cable from point A on one bank of an 800-foot wide, straight
river to point B on the opposite bank 1600 feet downstream. The power company will choose a
point C on the same side of the river as B, and will lay a cable underwater from A to C and on
land from C to B. It costs $3 a foot to lay the cable on land and $5 a foot to lay it underwater.
Where should point C be chosen to minimize the total cost and what is the minimum cost?
(11) A closed box with square base is to be built to house an ant colony. The bottom of the box
and all four sides will be made of material costing $1 per square foot, and the top is to be
constructed of glass costing $5 per square foot. What are the dimensions of the box of greatest
volume that can be constructed for $72?
(12) Explain the difference between the average rate of change in a function and the instantaneous
rate of change. What’s the difference in the way you compute these quantities? Illustrate by
considering the average rate of change of the function f (t) = t3 + 5t on the interval [1, 3] and
the instantaneous rate of change of the same function at t = 2.
(13) The function y = f (x) is graphed below.
(a) lim f (x) =
x→20
(b) lim
h→0
f (h)−10
h
=
(d) f 00 (10) =
(e) If g(x) = ex f (x), then g(0) =
(c) f 0 (10)=
(14) A weather balloon is rising vertically at the rate of 10 ft/s. An observer is standing on the
ground 300 ft horizontally from the point where the balloon was released.
(a) At what rate is the distance between the observer and the balloon changing when the balloon
is 400 ft high?
(b) At what rate is the angle of elevation of the observer’s line of sight to the balloon changing
when the balloon is 400 feet high?
(15) The right triangle shown below is undergoing a transformation so that the side marked x is
increasing at the rate of 3 cm/second and the side marked y is decreasing at the rate of 2
cm/second. How fast is the perimeter changing when x = 50 cm and y = 120 cm? Is the
perimeter increasing or decreasing at that instant?
(16) An object moves according to the equation s = t(2t − 1)3 , where s is the position of the object
in feet and t is time in seconds. Find the acceleration when t = 1.
(17) The graph of the derivative of a certain function f appears below.
(a)
(b)
(c)
(d)
(e)
Suppose f (1) = 5. Find an equation of the line tangent to the graph of f at (1, 5).
Suppose f (−1) = −2. Could f (3) = −6? Why or why not?
Estimate f 00 (−3).
At which values of x does f (x) have points of inflection?
At which value of x in the interval [−4, 5] does f (x) achieve its largest value? Its smallest
value?
(18) Some values of a function f and its derivative f 0 are tabulated below:
x
f (x)
f 0 (x)
-3 -2 -1 0 1 2 3
4 -1 1 3 -2 2 5
0 1 2 -1 3 -2 -3
(a) Let g(x) = 2f (x + 3). Evaluate g 0 (−2).
x
(b) Let h(x) = fln(x)
. Is x = 1 a critical number for h? Justify your answer.
4
(c) Let j(x) = f (x ). Is j increasing of decreasing at x = −1? Justify your answer.
(19) Given the curve x2 y − 5xy 3 + 6 = 0,
(a) Find the slope and equation of the tangent line to this curve at (3, 1).
(b) Use a linear approximation to estimate the y-value of a point on this curve with x-value
3.5.
a
(20) Let f (x) = x2 + .
x
(a) What value of a makes f have a local minimum at x = 2 ?
(b) What value of a makes f have a point of inflection at x = 1?
(c) Is there any value of a for which f can have a local maximum? Why or why not?
(21) You are designing right circular cylindrical cans with volumes of 1000 cubic centimeters. The
manufacturer of these cans will take waste into account. There is no waste in cutting the
aluminum for the sides, but the tops and bottoms of radius r will be cut from squares that
measure 2r cm on a side. The total amount of aluminum used up by each can will therefore
be A = 8r2 +2πrh. Find the values of r and h which will minimize the amount of aluminum used.
(22) Which of the graphs below suggest a function y = f (x) that is
(a) continuous for all real x?
(b) differentiable for all real x?
(c) both (a) and (b)?
(d) neither (a) nor (b)?
Explain your answer in each case.
(23) Let f (x) =
x2 + x + 1
. Answer the following questions:
x2
(a)
(b)
(c)
(d)
Determine all x- and y-intercepts for the curve.
Determine any vertical asymptotes and horizontal asymptotes for the curve.
Find all critical numbers for f (x).
Find the intervals on which f (x) is increasing, and the intervals on which f (x) is decreasing.
Determine x- and y-coordinates of all local minimum(s) and local maximum(s).
(e) Find the intervals on which f (x) is concave up and concave down. Find the x- and ycoordinates of all of the inflection points.
(f) Using (a)-(e), sketch the curve.
(24) Evaluate the following limits.
sin x − x cos x
(a) lim
x→0
x3
cos x
(b) lim+
x→0
x
(c) lim
3
x
sin
2
x
x→∞
1
(d) lim x x−1
x→1
ANSWERS
(1) (a) 17
(e) 4
(b) 13
(f) 2
(c) −∞
(g) 14
(d) +∞
(h) limit does not exist
(2) y = 1, x = 2.
(3) a = −2, a = 4.
(4) y = − 25 x +
17
.
5
(5) (a) 5(2x3 + 5x2 − 6x − 4)4 (6x2 + 10x − 6)
1
(c) 1/2
2x (3x + 1)3/2
x + 4xy
(e) − 2
2x − 12y 3
20
x
(g)
− 2
5x + 2) x + 5)
(6) (a) negative
(d) zero
(g) positive
(b) 15 tan2 (5x) sec2 (5x)
(d) e2x (cos x + 2 sin x)
(f)
3x2
x6 + 1
√
(h)(tan x)
(b) positive
(e) positive
(h) zero
x
√ sec2 x
1
√ · ln(tan x) + x ·
tan x
2 x
(c) positive
(f) negative
(i) negative
√
√
(7) Local maximum at x = −3 2, inflection points at x = −3, 0, 3, and local minimum at x = 3 2.
(8) Concave up for x < 2, concave down for x > 2.
(9) Min: −3, Max: 0.
(10) Go diagonally across the river to a point 600 ft downstream, then along land for 1000 ft. Min
cost: $8000.
(11) Base: 2 ft by 2 ft; height: 6 ft.
(12) vavg = 18, v = 17.
(13) (a) -10 (b) -1 (c) -1 (d) 0 (e) 9.
(14) (a) 8 ft/s.
(b)
3
250
radians/s.
(15) The perimeter is increasing at the rate of
4
13
cm/sec.
(16) 36 ft/s2 .
(17) (a)
(b)
(c)
(d)
y = (1.5)x + 3.5.
No, since f 0 (x) > 0 for x > −3, f (x) is always increasing on this interval.
f 00 (−3) ≈ 4.
at x = −1 and x = 3.
(e) largest value at x = 5, smallest value at x = −3.
(18) (a) 6
(b) No, h0 (1) 6= 0.
(c) j is decreasing because j 0 (−1) = −12 < 0.
(19) (a) Slope = 1/36, y =
(20) (a) 16.
(21) r = 5 cm, h =
(22) (a)
(b)
(c)
(d)
1
(x
36
(b) -1.
40
π
− 3) + 1 =
1
x
36
+
33
.
36
(b) y ≈
73
.
72
(c) No.
cm.
I and III.
III.
III.
II.
(23) (a)
(b)
(c)
(d)
No x or y-intercepts.
x = 0 is the only vertical asymptote, y = 1 is the horizontal asymptote.
x = −2, x = 0 are the only critical numbers.
f is decreasing on (−∞, −2) and (0, ∞); f is increasing on (−2, 0). f has a local minimum
at (−2, 34 ).
(e) f is concave down on (−∞, −3); f is concave up on (−3, 0) and (0, ∞). The only inflection
point is at (−3, 97 ).
(24) (a)
1
3
(b) ∞
(c)
3
2
(d) e