Exam 3 Study Guide

MATH 122A · Fall 2014

Section 4.6

1. (This is example 6 on page 236.) An airplane is flying at a speed of 450 km/hr at a constantaltitude of 5 km. The airplane approaches a camera mounted on the ground. Let θ be the angleof elevation above the ground at which the camera is pointed. When θ = π/3, how fast doesthe camera have to rotate in order to keep the plane in view?

2. Gasoline is pouring into a vertical cylindrical tank of radius 3 feet. When the depth of thegasoline is 4 feet, the depth is increasing at 0.2 ft/s. How fast is the volume of gasoline changingat this instant.

2

3. A person 5 feet tall is walking toward an 18 foot pole. A light is positioned at the top of thepole. If the person is walking at a constant rate of 6 ft/s, find the rate at which the length ofthe person’s shadow is changing when the person is 30 feet from the pole.

Section 4.74. Determine whether the limit exists, and if possible, evaluate it.

(a)

limt→π

sin2 t

t− π

3

(b)limx→0+

x lnx

(c)

limx→0

1− cosh(3x)

x

(d)

limx→0

(1

x− 1

sinx

)

(e)

limt→0

sin2At

cosAt− 1, A 6= 0.

4

(f)limx→0+

xa lnx,

where a is a positive constant.

(g)

limx→∞

(1 + sin

(3

x

))x

5. Determine which function dominates as x→∞.(a) x5 and 0.1x7.

5

(b) ln(x+ 3) and x0.2.

(c) 0.01x3 and 50x2.

(d) x10 and e0.1x.

6

Section 5.16. A car, initially traveling at a speed of 50 ft/sec, brakes at a constant rate, coming to a stop in

5 seconds.(a) Graph the velocity function from t = 0 to t = 5.

(b) How far does the car travel during the period in which it is braking?

(c) How far does the car travel if its initial velocity is doubled, but it brakes at the sameconstant rate?

7. At time, t, in seconds, your velocity, v, in m/s, is given by

v(t) = 1 + t2 for 0 ≤ t ≤ 6.

Use ∆t = 2 to estimate the distance traveled during this time. Find the upper and lowerestimates, and then average the two.

7

Section 5.28. (a) Sketch a graph of y = sinx on the interval 0 ≤ x ≤ π/2. On the same graph, sketch the

rectangles corresponding to a left-hand Riemann sum with n = 6 subdivisions. What is thecorresponding value of ∆x? Write out the terms in the sum, and then use your calculatorto evaluate the sum.

(b) On another sketch of y = sinx on [0, π/2], sketch the rectangles corresponding to the right-hand Riemann sum approximation with n = 6. Write out the terms in the sum, and thenuse your calculator to evaluate the sum.

(c) Find a good n = 6 approximation for∫ π/20 sinxdx by taking the average of the two Riemann

sum approximations that you calculated above.

8

9. Use the table to estimate∫ 400 f(x)dx. What values of n and ∆x did you use?

x 0 10 20 30 40f(x) 350 410 435 450 460

10. Use the table to estimate∫ 150 f(x)dx.

x 0 3 6 9 12 15f(x) 50 48 44 36 24 8

11. Problem number 32 on page 288.

9

Section 5.312. Oil leaks out of a tanker at a rate of r = f(t) gallons per minute, where t is measured in minutes.

Write a definite integral expressing the total quantity of oil which leaks out of the tanker in thefirst hour.

13. If f(x) is measured in pounds (lbs), and x is measured in feet, what are the units of∫ ba f(x)dx?

14. Explain in words what the following integral represents:∫ 50 s(x)dx, where s(x) is the rate of

change of salinity (salt concentration) in gm/liter per cm in sea water, and where x is the depthbelow the surface of the water, in cm.

15. Pollution is removed from a lake on day t at a rate of f(t) kg/day.(a) Explain the meaning of the statement f(12) = 500.

(b) If∫ 155 f(t)dt = 4000, give the units of the 5, the 15, and the 4000.

10

(c) Give the meaning of∫ 155 f(t)dt = 4000.

Section 5.4

16. If f(x) is odd and∫ 3−2 f(x)dx = 30, find

∫ 32 f(x)dx.

17. If∫ 52 (2f(x) + 3)dx = 17, find

∫ 52 f(x)dx.

18. If f(x) is even and∫ 2−2(f(x)− 3)dx = 8, find

∫ 20 f(x)dx.

19. Without any computation, find ∫ π/4

−π/4x3 cos(x2)dx.

11

20. If the average value of f on the interval 2 ≤ x ≤ 5 is 4, find∫ 52 (3f(x) + 2)dx.

Section 6.1

For this section, I expect you to be able to take the graph of a function’s derivative, and fromit produce a reasonable sketch of the graph of the original function. Good examples of this areproblems 2 - 11 on page 323, and problems 16-20 on page 324.

21. Estimate f(x) for x = 2, 4, 6, using the given values of f ′(x) and the fact that f(0) = 100.

x 0 2 4 6f ′(x) 10 18 23 25

22. Estimate f(x) for x = 2, 4, 6, using the given values of f ′(x) and the fact that f(0) = 50.

x 0 2 4 6f ′(x) 17 15 10 2

12

Section 6.2

23. Find the following indefinite integrals(a)

∫(5x+ 7)dx

(b)∫

(2 + cos t)dt

(c)∫

(3ex + 2 sinx)dx

(d)∫

(5x2 + 2√x)dx

13

(e)∫ 8√

xdx

(f)∫

(ex + 5)dx

(g)∫ (√

x3 − 2

x

)dx

24. Find the following definite integrals exactly, using the Fundamental Theorem of Calculus. Thismeans do not write decimal approximations anywhere.

(a)∫ 30 (x2 + 4x+ 3)dx

14

(b)∫ π/40 sinxdx

(c)∫ 20 3exdx

(d)∫ 10 sin θdθ

(e)∫ 20

(x3

3+ 2x

)dx

15

25. Find the exact area of the region bounded by the x-axis and the graph of y = x3 − x.

26. Find the exact area between the curves y = x2 and y = 2− x2.

28. Consider the area between the curve y = ex − 2 and the x-axis, between x = 0 and x = c forc > 0. Find the value of c making the area above the axis equal to the area below the axis.

16

29. The average value of the function v(x) = 6/x2 on the interval [1, c] is equal to 1. Find the valueof c.

30. Find the exact area between the curves y = x2 and x = y2.

31. Find the exact area enclosed by the curve y = x2(1− x)2 and the x-axis.

17

Section 6.3

32. Find the general solution to the given differential equation.

(a)dy

dx= x3 − x

(b)dz

dt=√t

(c)dh

dz= z + ez

(d)dy

dx= 4x3 − 7

18

(e)dP

dt= 2 + sin t

(f)dy

dt=

7

cos2 t

33. Find the solution to the initial value problem.

(a)dy

dx=

2

1 + x2, y(1) = π/4.

(b)dy

dx= 2x, y(0) = 0.

19

(c)dy

dx= sinx, y(0) = 0

(d)dy

dx= ex, y(0) = 7.

Section 6.4

34. Find the following derivatives.

(a)d

dx

∫ x2 ln(t2 + 1)dt

(b)d

dx

∫ x20 ln(1 + t2)dt.

20

(c)d

dt

∫ sin t1 cos(x2)dx.

(d)d

dt

∫ 42t sin(

√x)dx.

(e)d

dx

∫ x2−x2 e

t2dt.

(f)d

dx

∫ x3x e−t

2dt.

21

35. Evaluate the following limits:(a)

limx→0

∫ x0 sin(2t)dt∫ x0 cos(t2)dt

(b)

limt→0

∫ t0

√1 + x2dx∫ t

0 cos2 xdx