Calculus Review

Basic Derivatives

100

What is the derivative of y = 3x² ?

y' = 6x

200

Find the derivative of f(x) = 5

0

300

Find the derivative of
f(x) = 6x³ + x - 2

18x² + 1

400

Find the derivative of
f(x) = 3(x+5)^1/2

3*1/2*(x+5)^-1/2

Definitions

100

If F'=f, then F is a(n) _____ of f.

Antiderivative (not integral!)

200

If f'(c)=0, then c is a(n) _________ of f(x).

Critical number

300

If f''(c)>0, then (c,f(c)) is a local _________ of f(x).

Minimum

400

In the expression ∫f(x)dx = F(x)+C, what is C called?

The constant of integration.

Chain Rule

100

Find y' if y = (3x - 9)⁴

12*(3x - 9)³

200

Find the derivative of
f(x) = (3x+5)^1/2

1/2*(3x+5)^-1/2*3

300

Find y' if y = (9x² + 4)^1/3

(6x)/(9x² + 4)^2/3

400

If f(x) = 3*(x²-2x)^-1/3, find f'(x)

3*-1/3*(x²-2x)^-4/3*(2x-2)

Polynomials

100

Find the indefinite integral
y = x² + 2x³

1/3*x ³+2/4*x⁴+C

200

Find any antiderivative of
y = x³ * (2x+1)

2/5*x⁵+1/4*x⁴

300

Find the indefinite integral of
f(x) = 4x^1/3-5/x

3*x^4/3-5*ln(x)+C

400

Find any antiderivative of
g(x) = (x+1)³

1/4*(x+1)⁴

Substitution

100

Find any antiderivative of
y = (16x+2)*(4x²+x+1)^1/3

2*3/4*(4x²+x+1)^4/3

200

If f(t)= 3(2t+1)^-2, find the indefinite integral of f(t).

-3/2*(2t+1)^-1+C

300

Evaluate
2 ∫ (x+1)(x²+2x)^-1 dx

ln(x²+2x)+C

400

Compute
∫[(2x+1)³+2(2x+1)][2x+1]^-1 dx

1/2*[(2x+1)²+2(2x+1)]+C