AP Calculus Review Flashcards

Flashcards covering derivatives, integration, theorems, and applications in AP Calculus.

What condition must y=f(x) satisfy at critical points for derivatives?

y=f(x) must be continuous.

How is a local minimum identified using the first derivative?

dy/dx goes from (-, 0, +) or (-, undefined, +) or dy/dx > 0.

How is a local maximum identified using the first derivative?

dy/dx goes from (+, 0, -) or (+, undefined, -) or dy/dx < 0.

How is a point of inflection identified using concavity?

Concavity changes: goes from (+, 0, -), (-, 0, +), (+, undefined, -), or (-, undefined, +).

State the Chain Rule for differentiation.

d/dx [f(u)] = f'(u) * du/dx OR dy/dx = (dy/du) * (du/dx)

State the Product Rule for differentiation.

d/dx (uv) = u'v + uv'

State the Quotient Rule for differentiation.

d/dx (u/v) = (u'v - uv') / v^2

What is the derivative of sin(x)?

cos(x)

What is the derivative of cos(x)?

-sin(x)

What is the derivative of tan(x)?

sec²(x)

What is the derivative of cot(x)?

-csc²(x)

What is the derivative of sec(x)?

sec(x)tan(x)

What is the derivative of csc(x)?

-csc(x)cot(x)

What is the derivative of ln(u)?

(1/u) * (du/dx)

What is the derivative of e^u?

e^u * (du/dx)

State the Fundamental Theorem of Calculus.

∫[a, b] f(x) dx = F(b) - F(a), where F'(x) = f(x)

State the corollary to the Fundamental Theorem of Calculus.

d/dx ∫[a(x), b(x)] f(t) dt = f(b(x))b'(x) - f(a(x))a'(x)

State the Intermediate Value Theorem.

If f(x) is continuous on [a, b], and y is a number between f(a) and f(b), then there exists at least one number x=C in (a, b) such that f(c)=y.

State the Mean Value Theorem.

If f(x) is continuous on [a, b], AND the first derivative exists on (a, b), then there is at least one number x = c in (a, b) such that f'(c) = (f(b)-f(a))/(b-a)

Area between curves is calculated with what integral?

Integral of (top - bottom) dx OR Integral of (right - left) dy

State the formula for Arc Length.

L = ∫ √[1 + (f'(x))²] dx

How is velocity found from position?

velocity = d/dt (position)

How is acceleration found from velocity?

acceleration = d/dt (velocity)

Define average velocity.

Average velocity = (final position - initial position) / (total time)