AP Calculus Review - Differentiation

AP Calculus Review - Differentiation

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Section 1

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Derivative of cot(x)

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Cards (65)

Section 1

(50 cards)

Derivative of cot(x)

Front

Back

Derivative of sin(x)

Front

Back

Derivative of f(g(x)) Commonly referred to as the "Chain Rule"

Front

Back

If f is differentiable at x=a, then...

Front

... this limit exists

Back

If f"(x) > 0 on (a, b), then...

Front

... the graph of f is concave upward on (a, b)

Back

The 2nd Derivative Test f'(a) = 0 and f"(a) < 0

Front

... then f has a maximum at x = a

Back

Derivative of sec(x)

Front

Back

Derivative of a Generic Logarithmic function

Front

Back

f has a local (relative) minimum at x = a

Front

f(a) is less than or equal to every other y-value in some interval containing x = a.

Back

dx/dy = 0

Front

The graph of f has a vertical tangent.

Back

If f'(x) < 0 on (a, b), then...

Front

... f is decreasing on (a, b)

Back

At x = a, f is continuous and changes from increasing to decreasing...

Front

... then f has a maximum at x = a.

Back

Point of Inflection

Front

A Point of Inflection indicates a change in concavity. You find these inflections by first finding a candidate when the second derivative is equal to zero. Then check the value of the second derivative to the left and right of each candidate - if there is a change in sign from left to right, then the candidate is a point of inflection.

Back

Critical Points of f

Front

Critical Points occur: > at endpoints of the domain > where the derivative of f does not exist > where the derivative of f equals zero

Back

If f is differentiable at x=a ...

Front

then f is continuous at x=a.

Back

f has a local (relative) maximum at x = a

Front

f(a) is greater than or equal to every other y-value in some interval containing x = a.

Back

Rolle's Theorem

Front

Back

Derivative Product Rule

Front

Back

Derivative of sin⁻¹(u) with respect to x

Front

Back

Mean Value Theorem (by definition)

Front

Back

What identity do you use to find the derivative of cos⁻¹(x)?

Front

Back

The 2nd Derivative Test f'(a) = 0 and f"(a) > 0

Front

... then f has a minimum at x = a

Back

Derivative of cos(x)

Front

Back

Definition of a Derivative f'(x)

Front

Back

Speed

Front

where v(t) is velocity

Back

Definition of a Derivative at a Point f'(a)

Front

Back

At x = a, f is continuous and changes from decreasing to increasing...

Front

... then f has a minimum at x = a.

Back

If f'(x) > 0 on (a, b), then...

Front

... f is increasing on (a, b)

Back

If f"(x) < 0 on (a, b), then...

Front

... the graph of f is concave downward on (a, b)

Back

If f'(x) is decreasing on (a, b)

Front

... the graph of f is concave downward on (a, b)

Back

What identity do you use to find the derivative of csc⁻¹(x)?

Front

Back

Average Rate of Change of f on [a, b]

Front

Back

Derivative of tan⁻¹(u) with respect to x

Front

Back

Derivative of an exponential function

Front

Back

Derivative of sec⁻¹(u) with respect to x

Front

Back

Mean Value Theorem (geometrically)

Front

Back

Derivative of the Natural Log function

Front

Back

f'(a) = 0

Front

The graph of f has a horizontal tangent at x=a.

Back

Instantaneous Rate of Change of f at x=a

Front

f'(a)

Back

Derivative of xⁿ

Front

Back

Velocity

Front

where s(t) is the position

Back

Derivative of csc(x)

Front

Back

If f'(x) is increasing on (a, b)

Front

... the graph of f is concave upward on (a, b)

Back

Derivative Quotient Rule

Front

Back

What identity do you use to find the derivative of cot⁻¹(x)?

Front

Back

Derivative of an Inverse Function

Front

Back

f has a global (absolute) maximum at x = a

Front

f(a) is greater than or equal to every other y-value of f.

Back

f has a global (absolute) minimum at x = a

Front

f(a) is less than or equal to every other y-value of f.

Back

Derivative of tan(x)

Front

Back

Derivative of the natural base function

Front

Back

Section 2

(15 cards)

Increasing Speed

Front

Velocity and Acceleration have the same sign!

Back

Decreasing Speed

Front

Velocity and Acceleration have opposite signs!

Back

Fill in the blank! If the graph of f is concave upward on [a, b], then the linear approximation to f at x = a ________________ f(b).

Front

underestimates

Back

Linear Approximation to f at x = a

Front

Back

Acceleration (in terms of velocity and position)

Front

Back

Derivative of Parametric Equations

Front

Back

The Normal Line to a curve at x = a is...

Front

... perpendicular to the Tangent Line to the curve at x = a

Back

An object in motion along a line reverses direction when...

Front

... the sign of the object's velocity changes!

Back

If f has a vertical tangent at x = a...

Front

... then, while f is continuous at x = a, f is not differentiable at x = a

Back

Corner

Front

A function is continuous at a corner, but it's not differentiable there!

Back

The slope of the Tangent Line to a curve (f) at x = a

Front

f'(a)

Back

Cusp

Front

A cusp is where a curve meets a straight line. A function is continuous at a cusp, but it's not differentiable there!

Back

Fill in the blank! If the graph of f is concave downward on [a, b], then the linear approximation to f at x = a ________________ f(b).

Front

overestimates

Back

An object is at rest when...

Front

... the velocity is equal to zero. v(t) = 0

Back

Second Derivative of a Parametric Equation

Front

Back