AP Calculus AB Review

Testing Your Limits

100

lim x→0 (3^x-1)/x

ln3

200

Is this function differentiable at x=3?

f(x)=3x²-30, x ≥ 3

f(x)=x³-10x, x < 3

No

300

lim x→∞ (3x⁵-6x)/e^x

0

400

What type of discontinuity?

g(x)=x²-5, x < 3

g(x)=9-2x, x ≥ 3

Jump Discontinuity

500

lim x→0 sin(3x)/(2-√(4-x))

12

Devious Derivatives

100

Find f'(x)

f(x)=(4x+5)³

12(4X+5)²

200

d/dx sec^-1(3x)

3/(|3x|√(9x² - 1))

300

Approximate f(2.2) using the line tangent to f(x)=√(2+x) at x=2

2.05 OR 41/20

400

If f(x)=√(x²-4) and g(x)=3x-2, then the derivative of f(g(x)) at x=3 is

7/√5 or (7√5)/5

500

Let f(x)=(2x+1)³, and let g(x) be the inverse function of f. Given that f(0)=1, what is g'(1)?

1/6

Explicitly Implicit

100

find dy/dx given x²+y²=9

-x/y

200

If x²+xy+y³=0, then, in terms of x and y, dy/dx is

-(2x+y)/(x+3y²)

300

If tan(xy)=x, then dy/dx is

(cos²(xy)-y)/x OR (1-ysec²(xy))/(xsec²(xy))

400

If x+2xy-y²=2, then at the point (1, 1), dy/dx is

Does not exist at (1,1)

500

If cos(x²)=xe^y, then, dy/dx is

(-2xsin(x²)-e^y)/(xe^y)

Irregular Integration

100

∫(x²+1)²dx=

1/5x⁵+2/3x³+x+C

200

∫ 0 to π of (cosx+1)dx

π

300

∫sec²xdx

tanx+C

400

∫ 2 to 3 of x(x²+1)dx

75/4

500

If ∫ 1 to 10 of f(x)dx=4 and ∫ 10 to 3 of f(x)dx=7, then ∫ 1 to 3 of f(x) is

11

What Could Possibly Be Here?

100

If lim x→3 f(x)=7, which of the following must be true? 

I. f is continuous at x=3

II. f is differentiable at x=3

III f(3)=7

None because f(3) is not stated

200

lim n→∞ (3n³-5n)/(n³-2n²+1)

3

300

The acceleration of a particle moving along the x-axis at time t is given by a(t)=6t-2. If the velocity is 25 when t=3 and the position is 10 when t=1, then the position x(t)=

x(t)=t³-t²+4t+6

400

For what value of x does the function f'(x)=(x-2)(x-3)² have a relative maximum?

7/3

500

A particle moves along a line so that at time t, where 0≤t≤π, its position is given by s(t)=-4cos(t)-(t²/2)+10. What is the velocity of the particle when its acceleration is zero?

v(1.31812)=2.55487