AP Calculus Review

Derivatives

200

Find the derivative of x²+4x+3

What is 2x+4?

400

Find the derivative of -x³+4x²-5

What is -3x²+8x?

600

Find the second derivative of f(x)=sin(2x³-9x)

What is 12cosx(2x³-9x)-(6x²-9)²sin(2x³-9x)

800

Find the third derivative of f(x)=11x⁵-3x⁴+9x³+6x²

What is 660x²-72x+54?

1000

Find f'(3) when f(x)=(4x²+9)sin(7x³)

What is 7,457.258?

Antiderivatives

200

Find the antiderivative of 12-x

What is 12x-(x²/2)?

400

Find the antiderivative of 8x³-9x²+4

What is 2x⁴-3x³+4x?

600

Find the antiderivative of (√x)+(1/√x)

What is (2/3)x^3/2+x^1/2?

800

Find the antiderivative of (x²+2x-3)/x⁴

What is -(1/x)-(1/x²)+(1/x³)?

1000

Find the antiderivative of (3/t)-(7/t²)

What is 3ln|t|+(2/t)?

Theorems

200

Using this theorem, show that lim_t→0(t²sin(1/t)=0:

-1≤sin(1/t)≤1

-t²≤t²sin(1/t)≤t²

lim_t→0(-t²)=0 and lim_t→0(t²)=0

Therefore, lim_t→0(t²sin(1/t)=0

What is the Squeeze Theorem?

400

This theorem states "if f is a differentiable function on the interval [a,b], then there exists a number c between a and b such that f'(c)=[f(b)-f(a)]/(b-a)."

What is the Mean Value Theorem?

600

This theorem states "if f is continuous on a closed interval [a,b], then f attains an absolute maximum value."

What is the Extreme Value Theorem?

800

This theorem states "if f is continuous on the interval [a,b], then ∫_a^b f(x)dx = f(b)-f(a), where f is the antiderivative of f, that is F'=f."

What is the Evaluation Theorem?

1000

This theorem states "if f has a local minimum or maximum and if f'(c) exists, then f'(c)=0.

What is Fermat's theorem?

Limit Laws

200

Solve this limit using variables: Lim_x→a C = 

What is C?

(Constant rule)

400

Solve this limit using variables: Lim_x→a xⁿ =

What is aⁿ?

(Power rule)

600

If lim_x→0 g(x)=-4 and lim_x→0 h(x)= -1,

Solve lim_x→0 √g(x)h(x)

What is 2?

800

If lim_x→0 f(x)=6 and lim_x→0 h(x)= -1,  

Solve lim_x→0[f(x)+h(x)]³

What is 125?

1000

If lim_x→0 g(x)=-4

Solve lim_x→0 ³√11+[g(x)]²

What is 3?

True/False

200

True or False:

When taking a derivative, (d/dx)cosx=-sinx

What is true?

400

True or false:

The chain rule states that f'(x)=g(h(x))=g'(h(x))h(x)

What is false?

(g'(h(x))h'(x)

600

True or false:

We cannot find the exact integral when we don't know the antiderivative for f

What is true?

800

True or false:

The integral of 4x⁶-2x³+7x-4 dx is (4/7)x⁷-(1/2)x⁴+(7/2)x²-4x

What is false?

(The answer is missing the constant, C)

1000

True or false?

Velocity is the antiderivative of position in terms of time.

What is false?