AP Calculus AB

Midterm Review

xy−y=2x+ 4If , then

dy

dxis

A.

B.

C.

D.

y−2x−1

2−yx−1

2

x−1

y−2x+1

x→ olim

sin3x5x

=

A.)3

5

B.)5

3C.)3

D.)5

E.)DNE

y =e2x + tan2x y '(π ) =If , then

A.)2e2π

B.)e2π +1

C.)2e2π + 2

D.)2eπ −2E.)0

Write the equation of the line tangent to y =ex+1at x =0

A.)y =ex+ eB.)y=xC.)y=x+1D.)y=x+ eE.)y=ex+1

The graph of y =ln(1−x)x+1 has vertical asymptote(s) at

A.)x =1B.)x=0C.)x=±1D.)x=−1E.)none

x→ 0lim

cos(π4

+ x)−cosπ4

x=

A.)−32

B.)−22

C.)0

D.)22

E.)1

For what values of x is f (x)=2x3 −x2 + 2x concave up?

A.)x <16

B.)x< 0C.)x> 0

D.)x>16

E.)x> 6

If f(1)=2 and f’(1)=5, use the equation of the tangent line to the graph of f(x) at x=1 to approximate f(1.2).

A.)1

B.)1.2

C.)3

D.)5.4

E.)9

The graph of f ‘(x) is shown at right.Which of the following could be the graph of f (x)?

A.

B.

C.

D.

E.

f ‘(x)

Which of the following statements is true about thefigure at right?

-3-7 5

A. exists

B.

x→ −7limf (x)

x→ 5limf(x) exists

C.x→ −7limf(x) = f(7)

D.x→ 5limf(x) = f(5)

E. f (5)− f(−7)5 −−7

= f '(c)

How many points of inflection are there for the function

y =x+ cos2x on the interval [0,π ] ?

A. 0B. 1C. 2D. 3E. 4

GRAPHING CALCULATOR ALLOWED

Consider the function y =x3 −x2 −1 .

For what value(s) of x is the slope of the tangent line equal to 5?

A.)−1

B.)53

C.)−1and53

D.)13

E.)2.219

A pebble thrown into a pond creates circular ripples such thatthe rate of change of the radius is 6 cm/sec.How fast is the area of the ripple changing when the circumference is cm?12π

A.)6πB.)2πC.)12πD.)36πE.)6

sq.cm./sec.

sq.cm./sec.

sq.cm./sec.

sq.cm./sec.

sq.cm./sec.

GRAPHING CALCULATOR ALLOWED

Find the average rate of change of f (x)=secxon the interval 0,

π3

⎡⎣⎢

⎤⎦⎥

.

A. 0.396B. 0.955C. 1.350D. 1.910E. Undefined

In the figure shown at right,which of the following is true?

x→ 1limf(x) =3

x→ 1+limf(x) =3

f '(1)=1f '(1)=3The average rate of change of f(x) on [1,3] equals f '(2)

A.

B.

C.

D.

E.

A function f(x) is continuous on [a,b]. Which of the followingmust be true?

A. f has a maximum on [a,b]

B. f has a point of inflection on [a,b]

C. f '(c)=f(b)− f(a)

b−afor at least one c in the interval [a,b]

D. f '(c)=0 for at least one c in the interval [a,b]

E. f has a critical value on the interval (a,b)

If v(t)=ln(t2 + t+1) , then a(1)=

A.)1

3

B.)2

3C.)1

D.)4

3E.)3

CHALLENGE!

d

dx h→ 0lim

ln(x+h)−lnxh

⎛⎝⎜

⎞⎠⎟=

A.)−1x2

B.)1x

C.)−1D.)0E.)undefined