Calculus Review Flashcards

Flashcards to help review key vocabulary and theorems from Calculus lecture notes.

Relative Maximum

f(c) is a relative maximum if there exists δ > 0 such that for all x ∈ (c − δ, c + δ), f(c) ≥ f(x).

Relative Minimum

f(c) is a relative minimum if there exists δ > 0 such that for all x ∈ (c − δ, c + δ), f(c) ≤ f(x).

Relative Extremum

If either a relative maximum or minimum holds, f(c) is called a relative extremum.

Theorem 4.1

If f is differentiable at c and f(c) is a relative extremum, then f'(c) = 0.

Critical Value

A number c in the domain of f is a critical value if f'(c) = 0 or f'(c) does not exist.

Absolute Maximum

A number M is an absolute maximum of f on an interval Q if there exists c ∈ Q such that f(c) = M and f(x) ≤ M for all x ∈ Q.

Absolute Minimum

m is an absolute minimum if f(c) = m and f(x) ≥ m for all x ∈ Q.

Extremum Value Theorem

If f is continuous on [a, b], then f has both an absolute maximum and minimum on [a, b].

Increasing Function

If f(a) ≤ f(b) for all a, b ∈ Q such that a < b, then f is said to be an increasing function on Q.

Strictly Increasing Function

If f(a) < f(b) for all a, b ∈ Q such that a < b, then f is said to be a strictly increasing function on Q.

Decreasing Function

If f(a) ≥ f(b) for all a, b ∈ Q such that a < b, then f is said to be a decreasing function on Q.

Strictly Decreasing Function

If f(a) > f(b) for all a, b ∈ Q such that a < b, then f is said to be a strictly decreasing function on Q.

Theorem 4.3

Let the function f be continuous on [a, b] and differentiable on the open interval (a, b), then if f'(x) ≥ 0 for all x ∈ (a, b), then f is increasing on [a, b].

Theorem 4.3

Let the function f be continuous on [a, b] and differentiable on the open interval (a, b), then if f'(x) ≤ 0 for all x ∈ (a, b), then f is decreasing on [a, b].

Step 1 of Determining Relative Extremum Value/s (Using first derivative test)

Solve for f'(x).

Step 2 of Determining Relative Extremum Value/s (Using first derivative test)

Solve for the critical value/s of f(x).

Step 3 of Determining Relative Extremum Value/s (Using first derivative test)

Apply the first derivative test to the critical value/s.

Concave Upward

The graph of f is concave upward at (c, f(c)) if f′(c) exists and the graph of f lies above the tangent line near c.

Concave Downward

The graph of f is concave downward at (c, f(c)) if f′(c) exists and the graph of f lies below the tangent line near c.

Theorem 4.5

If f''(c) > 0, then the graph of f is concave upward at (c, f(c)).

Theorem 4.5

If f''(c) < 0, then the graph of f is concave downward at (c, f(c)).

Second Derivative Test

Let c be a critical number of a function f such that f'(c) = 0 and suppose f'' exist for all values of x in some open interval containing c, then if f''(c) < 0, then f has a relative maximum at c.

Second Derivative Test

Let c be a critical number of a function f such that f'(c) = 0 and suppose f'' exist for all values of x in some open interval containing c, then if f''(c) > 0, then f has a relative minimum at c.

Point of Inflection

The point (c, f(c)) is a point of inflection of the graph of f if the graph has a tangent line at (c, f(c)), and if there exists an open interval Q containing c such that if x ∈ Q, then one of the following holds: i. f''(x) < 0 if x < c, and f''(x) > 0 if x > c; or ii. f''(x) > 0 if x < c, and f''(x) < 0 if x > c.

Points of Inflection

Points of inflection are points where a graph changes its concavity.

Theorem 4.7

If f''(c) exists, then f''(c) = 0.

Sketching Curves - Step 1

Plot the relative extremum value/s (if any).

Sketching Curves - Step 2

Plot point/s of inflection (if any).

Optimization

Maximizing or minimizing some function relative to some set, often representing a range of choices available in a certain situation.