maths Calculus

Calculus II Homework #1 Fall 2015 Dr. S. Farmer, Instructor

Directions: (10 pts. each) Answer each of the following 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 explanations that sufficiently support your answers. Unless otherwise stated, you're to give the EXAXCT VALUES of answers instead of decimal approximations. In order to receive ANY credit for any applied/word problem, you MUST declare a variable (unless the variable(s) have already been declared in the problem) and set up and solve an appropriate mathematical expression that can be used to answer the question. Proper units must also be included in answers to applied problems. NO CREDIT WILL BE GIVEN FOR EITHER GUESSING OR CHECKING POSSIBLE ANSWERS WITHOUT SOLVING THE PROBLEM.

Finally, write ONLY FINAL ANSWERS ON THESE PAGES; you must show your work both according to homework guidelines and on YOUR OWN PAPER.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Solve the problem. 1) Does the graph of the function y = 2x + 4 sin x have any horizontal tangents in the interval

0 x 2 ? If so, where? 1)

Use implicit differentiation to find dy/dx.

2) x + y x - y

= x2 + y2 2)

Find dy dx

.

3) ln y = ey cos 8x 3)

Use implicit differentiation to find dy/dx and d2y/dx2. 4) x2 - y3 = 5 4)

Solve the problem. 5) At the two points where the curve x2 + 2xy + y2 = 9 crosses the x-axis, the tangents to the

curve are parallel. What is the common slope of these tangents? 5)

6) Find the normal to the curve x2 + y2 = 2x + 2y that is parallel to the line y + x = 0. 6)

Find the derivative of y with respect to x, t, or , as appropriate.

7) y = ln 1 + x

x4 7)

Use logarithmic differentiation to find the derivative of y with respect to the independent variable. 8) y = (cos x)x 8)

Find the derivative of y with respect to x. 9) y = tan-1 3x 9)

1

Solve the problem. Round your answer, if appropriate. 10) Electrical systems are governed by Ohm's law, which states that V = IR, where V = voltage,

I = current, and R = resistance. If the current in an electrical system is decreasing at a rate of 8 A/s while the voltage remains constant at 30 V, at what rate is the resistance increasing (in /sec) when the current is 52 A? (Do not round your answer.)

10)

11) The radius of a right circular cylinder is increasing at the rate of 5 in./sec, while the height is decreasing at the rate of 3 in./sec. At what rate is the volume of the cylinder changing when the radius is 14 in. and the height is 11 in.?

11)

Find the derivative at each critical point and determine the local extreme values. 12) y = x2/3(x2 - 16); x 0 12)

Solve the problem. On your paper you must JUSTIFY your answers using mathematical methods. 13) Let f(x) = x3 - 16x

(a) Does f (0) exist? (b) Does f (4) exist? (c) Does f (-4) exist? (d) Determine all extrema of f.

13)

Identify the function's extreme values in the given domain, and say where they are assumed. Tell which of the extreme values, if any, are absolute.

14) g(t) = t3 3

- 2.5t2 + 4t, 0 t < 14)

For the functions below, identify all local extrema and inflection points. You may sketch the graph electronically to verify, but not justify, your results.

15) y = x + sin x, 0 x 2 15)

2

16) y = x + cos 2x, 0 x 16)

Solve the problem.

17) Let f(x) = 1 2

x3- x2- 2x + 2.

(a) Find the intervals on which the function is increasing. (b) Find the intervals on which the function is decreasing. (c) Sketch a graph of y = f(x) along with the line through

(-2, f(-2)) and (0, f(0)). (d) Find any values of c in the interval (-2, 0) that satisfy

f (c) = f(0) - f(-2)

0 - (-2)

17)

18) What can you say about the inflection points of the quartic curve y = ax4 + b x3 + cx2 + dx + e, a 0? Give reasons for your answer.

18)

19) The accompanying figure shows a portion of the graph of a function that is twice-differentiable at all x except at x = p. At each of the labeled points, classify y and y as positive, negative, or zero.

19)

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20) A private shipping company will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed 108 in. What dimensions will give a box with a square end the largest possible volume?

20)

21) A rectangular sheet of perimeter 30 cm and dimensions x cm by y cm is to be rolled into a cylinder as shown in part (a) of the figure. What values of x and y give the largest volume?

21)

22) 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?

22)

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23) A solar station is to be built at ground level on the east-west line between two buildings as shown below. The distance between the two buildings is 100 meters, the height of the taller building is 80 meters, and the height of the smaller building is 40 meters. How far from the taller building should the station be placed in order to maximize the number of hours it will be in the sun on a day when the sun passes directly overhead? Note that

= - cot-1 x 80

- cot-1 100 - x

40

and find the value of x which maximizes .

23)

24) If the price charged for a candy bar is p(x) cents, then x thousand candy bars will be sold in

a certain city, where p(x) = 125 - x 14

. How many candy bars must be sold to maximize

revenue?

24)

25) A rectangular field is to be enclosed on four sides with a fence. Fencing costs $3 per foot for two opposite sides, and $8 per foot for the other two sides. Find the dimensions of the field of area 630 ft2 that would be the cheapest to enclose.

25)

Find the most general antiderivative.

26) 1 x3

- x3 - 1 5

dx 26)

27) x x + x

x2 dx 27)

Solve the initial value problem.

28) ds dt

= cos t - sin t, s 2

= 10 28)

29) dy dx

= 1

x + 6 , y(-5) = 3 29)

Solve the problem. 30) Given the acceleration, initial velocity, and initial position of a body moving along a

coordinate line at time t, find the body's position at time t. a = 18, v(0) = -7, s(0) = 3

30)

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