VOCABULARY. Define each term or phrase in the space ...Write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x. 47) y = cos 6 x Find dy/dx by implicit

Example Sheet

VOCABULARY. Define each term or phrase in the space provided or on a separate sheet of paper.

Find the average velocity of the function over the given interval.1) y = x2 + 8x, [2, 8]

Use the table to find the instantaneous velocity of y at the specified value of x.2) x = 1.

x y00.20.40.60.81.01.21.4

00.020.080.180.320.50.720.98

Find the slope of the curve for the given value of x.3) y = x2 + 11x - 15, x = 1

Use the graph to evaluate the limit.4) lim

x -1f(x)

1

5) limx 0

f(x)

Give an appropriate answer.6) Let lim

x 7f(x) = -10 and lim

x 7g(x) = 3. Find lim

x 7[f(x) · g(x)].

Find the limit.

7) limx 0

1 + x - 1x

Determine the limit by sketching an appropriate graph.

8) limx -2+

f(x), where f(x) = x2 + 6 for x -20 for x = -2

Find the limit, if it exists.

9) limx 8

8 - x8 - x

10) limx 2

x2 + 4x - 12x2 - 4

2

11) limx -6

x2 + 12x + 36x + 6

Find the limit.12) lim

x ( /2)+tan x

13) limx -1-

5x2 - 1

14) limx 0+

x2 - 3x + 2x3 - x

Find all vertical asymptotes of the given function.

15) g(x) =3x

x - 5

3

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Choose the graph that represents the given function without using a graphing utility.

16) f(x) =x

x2 + x + 216)

A) B)

C) D)

4

17) f(x) =x2 - 1

x317)

A) B)

C) D)


Find the limit.

18) limx

cos 4xx

Divide numerator and denominator by the highest power of x in the denominator to find the limit.

19) limx

3x - 3x - 5

5x + x2/3 + 2

20) limx

5x + 2

2x2 + 1

Find all horizontal asymptotes of the given function, if any.

21) h(x) = 5 -2x

5

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

Sketch the graph of a function y = f(x) that satisfies the given conditions.22) f(0) = 0, lim

x ±f(x) = 0, lim

x 4-f(x) = - , lim

x -4+f(x) = - , lim

x 4+f(x) = ,

limx -4-

f(x) = .

22)


Find numbers a and b, or k, so that f is continuous at every point.23)

f(x) =

x2,

kx,

if x 5

if x > 5

Find the intervals on which the function is continuous.

24) y =2

x + 4-

x25

Find all points where the function is discontinuous.25)


Provide an appropriate response.

26) Use the Intermediate Value Theorem to prove that 5 sin x = x has a solution between2

and .

26)

6


Use the graph to find a > 0 such that for all x, 0 < x - x0 < f(x) - L < .27)

y =32

x + 1

2.7

2.5

2.3

0.9 1 1.1

NOT TO SCALE

f(x) =32

x + 1

x0 = 1L = 2.5

= 0.2

28)

y = x

1.661.411.16

1.3575 2 2.7675

NOT TO SCALE

f(x) = xx0 = 2

L = 2

=14


Prove the limit statement

29) limx 7

1x

=17

29)

30) limx 3

4x2 - 9x- 9x - 3

= 15 30)

7


Compare the right-hand and left-hand derivatives to determine whether or not the function is differentiable at the pointwhose coordinates are given.

31)

y =1x

y = -1

31)

A) Since limx -1+ f (x) = -1 while limx -1- f (x) = 0, f(x) is not differentiable at x = -1.

B) Since limx -1+ f (x) = 0 while limx -1- f (x) = 1, f(x) is not differentiable at x = -1.

C) Since limx -1+ f (x) = 0 while limx -1- f (x) = -1, f(x) is not differentiable at x = -1.

D) Since limx -1+ f (x) = 0 while limx -1- f (x) = 0, f(x) is differentiable at x = -1.


Estimate the slope of the curve at the indicated point.32)

8

Solve the problem.33) Use the following information to graph the function f over the closed interval [-5, 6].

i) The graph of f is made of closed line segments joined end to end.ii) The graph starts at the point (-5, 1).iii) The derivative of f is the step function in the figure shown here.

Graph the equation and its tangent.34) Graph y = x2 + 3x - 8 and the tangent to the curve at the point whose x-coordinate is 1.

9


The graph of a function is given. Choose the answer that represents the graph of its derivative.35) 35)

A) B)

C) D)

10

36) 36)

A) B)

C) D)


Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find thevalue of the indicated derivative.

37) u(2) = 10, u (2) = 3, v(2) = -3, v (2) = -5.ddx

(2u - 4v) at x = 2

Find the derivative.38) y = 15x-2 - 3x3 + 14x

Find y .39) y = (3x - 5)(4x3 - x2 + 1)

11

Find the second derivative.40) y = 7x3 - 6x2 + 8

Provide an appropriate response.41) The curves y = ax2 + b and y = 2x2 + cx have a common tangent line at the point (-1, 0). Find a, b, and c.

42) The curve y = ax2 + bx + c passes through the point (2, 8) and is tangent to the line y = 2x at the origin. Find a, b,and c.

43) Find an equation for the tangent to the curve y =10x

x2 + 1 at the point (1, 5).


44) Find a value of c that will make

f(x) =

sin2 4xx2

, x 0

c, x = 0continuous at x = 0.

44)

45) Graph y = - tan x and its derivative together on -2

,2

. Does the graph of y = -tan x

appear to have a largest slope? If so, what is it? If not, explain.

45)


Solve the problem.46) The position of a body moving on a coordinate line is given by s = t2 - 6t + 6, with s in meters and t in seconds.

When, if ever, during the interval 0 t 6 does the body change direction?

Write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x.47) y = cos6 x

Find dy/dx by implicit differentiation.48) x1/3 - y1/3 = 1

Use implicit differentiation to find dy/dx and d2y/dx2.49) xy - x + y = 2

At the given point, find the line that is normal to the curve at the given point.50) x6y6 = 64, normal at (2, 1)

At the given point, find the slope of the curve or the line that is tangent to the curve, as requested.51) x6y6 = 64, slope at (2, 1)

12


Provide an appropriate response.52) Consider the quartic function f(x) = ax4 + bx3 + cx2 + dx + e, a 0. Must this function have

at least one critical point? Give reasons for your answer. (Hint: Must f (x) = 0 for some x?)How many local extreme values can f have?

52)


Find the absolute extreme values of the function on the interval.53) f(x) = |x - 8|, 6 x 11

Find the extreme values of the function and where they occur.

54) f(x) =1

x2 + 1

Find the largest open interval where the function is changing as requested.

55) Increasing f(x) =14

x2 -12

x

56) Decreasing f(x) = x - 8

Use the graph of the function f(x) to locate the local extrema and identify the intervals where the function is concave upand concave down.

57)

13

Graph the equation. Include the coordinates of any local extreme points and inflection points.

58) y =x2

x2 + 5

Sketch the graph and show all local extrema and inflection points.59) f(x) = |x2 - 2x|

Solve the problem.60) 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) = 124 -x16

. How many candy bars must be sold to maximize revenue?

61) A rectangular field is to be enclosed on four sides with a fence. Fencing costs $4 per foot for two opposite sides,and $7 per foot for the other two sides. Find the dimensions of the field of area 740 ft2 that would be thecheapest to enclose.


Provide an appropriate response.62) A marathoner ran the 26.2 mile New York City Marathon in 2.3 hrs. Did the runner ever

exceed a speed of 9 miles per hour?62)

14


L'Hopital's rule does not help with the given limit. Find the limit some other way.

63) limx 0+

sec xtan x

64) lim/2-

csccot

Use l'Hopital's Rule to evaluate the limit.

65) limx 1

x3 - 7x2 + 6x - 1

Find the most general antiderivative.

66) 4t2 +t

10dt

15


Which of the graphs shows the solution of the given initial value problem?

67) dydx

= -4x, y = -3 when x = 1 67)

A) B)

C) D)


Use differentiation to determine whether the integral formula is correct.

68) 36(9x + 5)3 dx = (9x + 5)4 + C

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed.69) f(x) = 16 - x2 between x = -4 and x = 4 using the midpoint sum with two rectangles of equal width.

Express the sum in sigma notation.70) 2 + 3 + 4 + 5 + 6 + 7

Express the limit as a definite integral.

71) lim0

n

k = 14xk7 xk; [5, 11]

16

Graph the integrand and use geometry to evaluate the integral.

72)5

07x dx

Solve the problem.

73) Suppose that f is continuous and that4

-4f(z) dz = 0 and

7

-4f(z) dz = 4. Find -

7

46f(x) dx .

74) Suppose that-2

-5g(t) dt = -12. Find

-2

-5

g(x)-12

dx and-5

-2- g(t) dt .

Find the derivative.

75) ddt

sin t

0

125 - u2

du

Evaluate the integral.

76)/2

022 sin x dx

Find the average value of the function over the given interval.77) y = 3 - x2; [-5, 1]

Find the point(s) at which the given function equals its average value on the given interval.78) f(x) = x ; [0, 12]

Use the substitution formula to evaluate the integral.

79)2

/33 cos2 x sin x dx

80)1

0

6 r dr

16 + 3r2

17

Documents

VOCABULARY. Define each term or phrase in the space ...Write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x. 47) y = cos 6 x Find dy/dx by implicit