Fill in the blank: The limit definition of the derivative of f(x) is f'(x) = _________.

lim_{Δx→0} (f(x+Δx) - f(x)) / Δx

lim_{Δx→∞} (f(x+Δx) - f(x)) / Δx

lim_{Δx→0} (f(x) - f(x+Δx)) / Δx

lim_{Δx→0} (f(x+Δx) + f(x)) / Δx

Fill in the blank: The alternate definition of the derivative of f at x = c is f'(c) = _________.

lim_{x→c} (f(x) - f(c)) / (x - c)

lim_{h→0} (f(x+h) - f(x)) / h

lim_{x→0} (f(x) - f(0)) / (x - 0)

lim_{x→c} (f(c) - f(x)) / (c - x)

Fill in the blank: The definition of average rate of change is _________.

Δy/Δx = (f(b) - f(a)) / (b - a)

Δx/Δy = (f(a) - f(b)) / (a - b)

Δy/Δx = (f(a) + f(b)) / (b - a)

Δy/Δx = (f(b) + f(a)) / (b + a)

Rolle’s Theorem: If f is continuous on [a, b], differentiable on (a, b), and…

…f(a) = f(b), then there exists c ∈ (a, b) such that f'(c) = 0.

…f(a) ≠ f(b), then there exists c ∈ (a, b) such that f'(c) = 0.

…f(a) = f(b), then there exists c ∈ (a, b) such that f'(c) ≠ 0.

…f(a) = f(b), then there exists c ∈ (a, b) such that f(c) = 0.

Mean Value Theorem for Derivatives: If f is continuous on [a, b] and differentiable on (a, b), then… (complete the statement)

there exists at least one c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).

Extreme Value Theorem: If $$f$$ is continuous on a closed interval, then… (complete the statement)

…f must have both an absolute maximum and an absolute minimum on the interval.

…f must be differentiable everywhere on the interval.

…f must be increasing on the interval.

…f must have a local maximum at the endpoints of the interval.

Intermediate Value Theorem: If $$f$$ is continuous on [a, b], then… (complete the statement)

…f must take on every y-value between f(a) and f(b).

…f must be differentiable on [a, b].

…f must have a maximum at either a or b.

…f must be constant on [a, b].

If a function is differentiable at a point, then _________.

it must be continuous at that point. (Differentiability implies continuity.)

it must be discontinuous at that point.

it must be non-differentiable at that point.

it must have a jump at that point.

List four ways in which a function can fail to be differentiable at a point.

Discontinuity, Corner, Cusp, Vertical tangent line

Oscillation, Periodicity, Symmetry, Translation

Continuous everywhere, Smooth curve, No inflection, Horizontal tangent

Constant function, Linear function, Quadratic function, Cubic function

A critical number (also known as a critical point or critical value) of f(x) is _________.

a value of x in the domain of f at which either f'(x) = 0 or f'(x) does not exist.

a value of x where f(x) is always increasing.

a value of x where f(x) is undefined for all x.

a value of x where f''(x) = 0 always.

To find a point of inflection, _________.

look for where f'' changes signs, or, equivalently, where f' changes direction.

look for where f' is zero and f'' is positive.

look for where f is increasing and f' is negative.

look for where f'' is always positive.

To find extreme values of a function, look for where _________.

the derivative is zero or undefined

the second derivative is positive

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Practice Problem

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Easy

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CCSS

HSF.TF.A.2, HSF-IF.C.7B, 8.G.C.9

+15

Standards-aligned

Rebecca Walker

Used 1+ times

FREE Resource

1 / (sin θ × cos θ)

sin θ × cos θ

Tags

CCSS.HSF.TF.C.8

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

Fill in the blank: sec θ (in terms of sine and/or cosine) = ______

1 / cos θ

cos θ / sin θ

sin θ

1 / sin θ

Tags

CCSS.HSF.TF.A.2

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

Which function is pictured in the graph?

Graph of y = cos x

Graph of y = sin x

Graph of y = sec x

Graph of y = tan x

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Fill in the blank: sin(π/2) = ____

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Fill in the blank: cos(π/2) = ______

Fill in the blank: cos(3π/2) = ______

Fill in the blank: tan θ (in terms of sine and/or cosine) = ______

Fill in the blank: sec θ (in terms of sine and/or cosine) = ______

Which function is pictured in the graph?

Which function is pictured in the graph?

Which function is pictured in the graph?

Pick the equation graphed.

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