Calculus Review

Chain Rule

100

What is the chain rule formula?

(f(g(x)))' = f'(g(x)) * g'(x)

200

d/dx [x² - 3x]⁵

5[x² - 3x]⁴ * (2x - 3)

300

d/dx [sin(6x)]

6cos(6x)

400

d/dt [ln(x(t) - 2)]

x'(t)/(x(t) - 2)

500

d/dt [cos(x(t))]

-sin(x(t)) * x'(t)

Integrals/General Solution

100

What does 'C' mean in the solution to an integral?

constant of integration - we could add any number there and get the same answer

200

What is the difference between a definite and indefinite integral?

Definite integrals are evaluated over a certain region - usually called limits of integration.

300

What are the bounds of integration here?

∫₂²⁰ √[4x² + 25x + 11]

lower bound is 2; upper bound is 20

400

d/dt [x(t)] = 3

x(t) = 3t + c

500

d/dt [ln(x(t))] = 3

x(t) = e^{3t + c}

Particular Solution

100

How does a particular solution vary from a general solution?

A particular solution does not have a C. You use an initial value, or given value, to solve for C.

200

x' = 3t² + 2t

x(0) = 3

x = t³ + t² + 3

300

x' = 1/t

x(1) = 1

x = ln(t) + 1

400

x' = -x³

x(0) = 0

x(t) = √(1/2t)

500

x' = 2

x(0) = 0

x(t) = 2t

Integrals

100

∫e^2x dx

(1/2)e^2x + c

200

∫sinh(2x) dx

(1/2)cosh(2x) + c

300

∫e^-xx + e^-xx² dx

-e^-x(x²+3x+3) + c

400

∫_pi^2(pi) -cos(x) dx

0

500

∫₀¹e^{-x^2} dx

-e^{-t^2} + 1

Derivatives

100

What is the product rule formula?

d/dx f(x) * g(x) = f'(x)g(x) + f(x)g'(x)

200

f(x) = x¹⁰

f'(x) = 10x⁹

300

f(x) = sin(x)

f'(x) = cos(x)

400

d/dx [ln(x)]

1/x

500

d/dx [xcos(x)]

cos(x) - xsin(x)