University of the District of Columbia
Calculus II
Division of Sciences and Mathematics
Dr. Shurron Farmer, Instructor
Homework #1
DUE DATE: Tuesday, February 5, 2019 at the beginning of class
Directions: (10 pts. each) Answer each of the questions below. In order to receive ANY credit for a question, you must SHOW
YOUR WORK using proper notation and clear and concise logic. You're graded on both the accuracy of your answers AND
your ability to write clear explanations that sufficiently support your answers. NOTE: YOU CANNOT GET CREDIT FOR
MULTIPLE CHOICE QUESTIONS BY SOLELY CHECKING EACH OF THE ANSWERS TO SEE WHICH CHOICE IS THE
RIGHT ANSWER. IN OTHER WORDS, you must SOLVE EACH PROBLEM.
Regarding all applie/word problems: in order to receive ANY credit for such problems, you must JUSTIFY YOUR WORK on
your own paper. This includes: 1. declaring a variable (unless a variable has already been declared in the problem) and
setting up and solving an appropriate mathematical equation, inequality, or function to solve the problem.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Find y .
1) y =
1
tan(10x - 5)
5
1)
Use logarithmic differentiation to find the derivative of y.
3 (5x + 1)(x + 4)2
2) y =
(x 3 + 6)(x + 8)
2)
Use logarithmic differentiation to find the derivative of y with respect to the independent variable.
3) y = (x + 2) sin x
Find the derivative of y with respect to x.
12x + 7
4) y = -cos-1
3
3)
4)
5) For the function f(x) = x4 - 2x2 - x, f(-1) = 0. Compute (f-1 )'(0).
5)
Solve the problem.
6) The function y = cot x -
2 3
csc x has an absolute maximum value on the interval 0 < x <
3
6)
. Find it.
7) How close does the semicircle y =
to two decimal places.
16 - x2 come to the point (1,
4)? Round your answer
7)
8) A small frictionless cart, attached to the wall by a spring, is pulled 10 cm back from its rest
position and released at time t = 0 to roll back and forth for 4 sec. Its position at time t is
s = 1 - 10 cos t. What is the cart's maximum speed? When is the cart moving that fast?
What is the magnitude of of the acceleration then?
8)
1
Find the value or values of c that satisfy the equation
f(b) - f(a)
= f'(c)
b-a
in the conclusion of the Mean Value Theorem for the given function and interval.
3
3
,
]
Round to the nearest thousandth.
9) f(x) = tan-1 x, [3
3
Solve the problem.
10) A baseball team is trying to determine what price to charge for tickets. At a price of $10 per
ticket, it averages 45,000 people per game. For every increase of $1, it loses 5,000 people.
Every person at the game spends an average of $5 on concessions. What price per ticket
should be charged in order to maximize revenue?
Find
9)
10)
dy
.
dx
11) e6y = cos (2x +y)
11)
12) 2 x + y = yln 2
12)
Provide an appropriate response.
13) Find a value of c that will make
sin2 4x
,
x 0
x2
f(x) =
c,
x=0
13)
continuous at x = 0.
Find the function with the given derivative whose graph passes through the point P.
1
14) f (x) = + 8x, P(1, 8)
x
Solve the problem.
15) Does the graph of the function y = tan x - x have any horizontal tangents in the interval
0 x 2 ? If so, where?
16) Without graphing the function, find lim (2x x
4x2 - 5x) .
14)
15)
16)
17) Find the points on the curve x2 + y2 = 2x + 2y where the tangent is parallel to the x-axis.
17)
18) Without graphing the function, find the slant asymptotes of the function
2x6 - 5x4 + 6
f(x) =
.
3x5 - 5x4 + 4
18)
2
19) Without graphing the function, find the equation of the tangent line(s) to the graph of the
16 + sin x
function y =
.
3 + sin x
19)
20) The position of a particle moving along the x-axis is given by
x(t) = cos( t2 ) for 0 t 3.
20)
(a)
(b)
(c)
(d)
Give the particle's velocity as a function of t.
Give the particle's acceleration as a function of t.
For what values of t is the particle moving to the right?
Find the acceleration at the first instant when the particle
returns to its starting position.
Solve the problem. Round your answer to the nearest whole number, if appropriate.
21) Water is being drained from a container which has the shape of an inverted right circular
cone. The container has a radius of 6.00 inches at the top and a height of 10.0 inches. At the
instant when the water in the container is 9.00 inches deep, the surface level is falling at a
rate of 1.2 in./sec. Find the rate at which water is being drained from the container.
21)
22) A rocket is launched vertically upward from a point 2 miles west of an observer on the
ground. What is the speed of the rocket when the angle of elevation (from the horizontal)
of the observer's line of sight to the rocket is 50° and is increasing at 5° per second? Find
your answer in terms of miles per hour, and approximate your solution to the nearest
whole number.
22)
23) The center of a circular pond of radius 100 feet is 200 feet from the straight side of an
exisiting fence running along a freeway. A rancher wants to enclose a rectangular field
containing the pond by using 2000 feet of fencing for 3 sides of the field and the existing
fence along the freeway as the fourth side. To give proper access to the pond he does not
want the fence to be closer to the pond than 100 feet. What dimensions should he use for
the largest such field?
23)
24) Refer to the previous question. What dimensions should the rancher use for the smallest
such field?
24)
25) For the derivative f'(x) =
x+2
, determine the local exterma and intervals of increasing
2
x (x - 6)
and decreasing of the function y = f(x).
3
25)
Answer Key
Testname: CAL. 2 HW 1 - SPR. 2019
1) 40 sec2 (10x - 5) tan(10x - 5)
1 3 (5x + 1)(x + 4)2
5
2
3x2
1
+
2)
+
+
+8
3
3
5x
1
x
4
3
x
(x + 6)(x + 8)
x +6
3) (x + 2) sin x cos x ln (x + 2) +
4)
sin x
x+2
12
9 - (12x + 7)2
5)
6) y = - csc2 (x) +
cos(x) =
1
2 3
3
2 3
2 3
csc(x)cot(x) = csc(x)
cot(x) - csc(x) = 0
3
3
x=
2 3
cot(x) - csc(x) = 0
3
2
3
7) x = 1.76
31.42 cm/sec; t = 0.5 sec, 1.5 sec, 2.5 sec, 3.5 sec; acceleration is 0 cm/sec2
8) 10
9) ±0.320
10) $7.00
-2 sin (2x + y)
11)
6y
6e
+ sin (2x + y)
12)
2x + y
2 x + y + yln 2 - 1
13) c = 16
14) f(x) = ln x + 4x2 + 4
15) Yes, at x = 0, x = , x = 2
16)
17) (1, 1 + 2), (1, 1 - 2)
18)
19)
20) (a) v(t) = -2 t sin( t2 )
(b) a(t) = -2 sin( t2 ) - 4 2 t2 cos( t2 )
(c) 1 < t < 2, 3 < t < 2, 5 < t < 6,
(d) -8 2 -78.957
7