AP Calculus AB Unit 5 Study Guide: Analytical Applications of Differentiation

Derivative (f'(x))

The instantaneous rate of change of f(x) with respect to x; geometrically, the slope of the tangent line to y=f(x) at x.

Instantaneous rate of change

How fast a function’s output is changing at a specific input value (an “at that instant” rate).

Tangent line

A line that touches a curve at a point and has the same slope as the curve at that point.

Slope interpretation of f'(a)

f'(a) is the slope of the line tangent to y=f(x) at x=a.

Positive derivative (f'(a) > 0)

Indicates the tangent line slopes upward to the right at x=a, so the function is increasing there.

Negative derivative (f'(a) < 0)

Indicates the tangent line slopes downward to the right at x=a, so the function is decreasing there.

Zero derivative (f'(a) = 0)

Indicates a horizontal tangent line at x=a; this alone does not guarantee a maximum or minimum.

Units of a derivative

“Units of output per unit of input” (e.g., meters per second if position is meters and time is seconds).

Average rate of change

The slope of the secant line on [a,b]: (f(b)−f(a)) / (b−a).

Secant line

A line through two points on a graph; its slope gives the average rate of change over an interval.

Mean Value Theorem (MVT)

Links average and instantaneous rate of change by guaranteeing a point where f'(c) equals the secant slope on [a,b].

Derivative sign analysis

Using whether f'(x) is positive/negative/zero to describe where f increases, decreases, or has horizontal tangents.

Increasing on an interval

If f'(x) > 0 throughout an interval, then f is increasing on that interval.

Decreasing on an interval

If f'(x) < 0 throughout an interval, then f is decreasing on that interval.

Horizontal tangent

A tangent line with slope 0 (occurs where f'(x)=0, if the derivative exists).

Estimating f'(a) from a graph

Approximate f'(a) by estimating the slope of the tangent line to the graph of f at x=a.

Estimating f'(a) from a table

Approximate f'(a) using a nearby secant slope, ideally with points equally spaced around a.

Symmetric difference quotient

A table-based estimate: f'(a) ≈ (f(a+h)−f(a−h)) / (2h), using points on both sides of a.

Leibniz notation (dy/dx)

A common notation meaning “derivative of y with respect to x.”

Prime notation (y')

A notation for the derivative of y with respect to x.

Second derivative (f''(x))

The derivative of f'(x); it measures how the slope (first derivative) is changing.

Interpreting T'(5) = −2.3

At t=5 minutes, temperature is decreasing at 2.3 degrees Celsius per minute (a rate, not the temperature value).

Local maximum

At x=c, f(c) is greater than or equal to nearby values in some open interval around c.

Local minimum

At x=c, f(c) is less than or equal to nearby values in some open interval around c.

Absolute (global) maximum

The greatest value of f(x) anywhere on the entire domain or interval under consideration.

Absolute (global) minimum

The least value of f(x) anywhere on the entire domain or interval under consideration.

Critical number

An x-value in the domain of f where f'(x)=0 or where f'(x) does not exist.

Critical point

A point (c, f(c)) on the graph where c is a critical number (a key location to check for extrema).

Fermat’s Theorem (idea)

If f has a local max/min at x=c and f'(c) exists, then f'(c)=0.

Corner/cusp as a critical number

A point where f is defined but f'(x) does not exist (e.g., f(x)=|x| at x=0), which can still be a local extremum.

Endpoints in absolute-extrema problems

On a closed interval [a,b], endpoints a and b can be absolute extrema even though they are not critical numbers.

Extreme Value Theorem (EVT)

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

Closed interval

An interval [a,b] that includes its endpoints; required for EVT’s guarantee of absolute extrema.

Continuity (for EVT/MVT)

No holes, jumps, or asymptotes on the interval; a required hypothesis for EVT on [a,b] and for MVT on [a,b].

Closed Interval Method (Candidates Test)

To find absolute extrema on [a,b], evaluate f at endpoints and at all critical numbers in (a,b), then compare values.

Candidate points

The x-values you must test for absolute extrema on [a,b]: endpoints plus critical numbers in the interior.

Rolle’s Theorem

If f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then there exists c in (a,b) with f'(c)=0.

MVT hypotheses

For MVT to apply, f must be continuous on [a,b] and differentiable on (a,b).

MVT conclusion

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

Differentiability

The property of having a derivative; fails at corners, cusps, vertical tangents, or discontinuities.

First Derivative Test

Classifies a critical number c by sign change of f': +→− gives a local max, −→+ gives a local min, no change gives neither.

Sign chart

A chart that records where a derivative is positive, negative, or zero by testing intervals between critical numbers.

Monotonicity

Whether a function is increasing or decreasing on an interval, often determined using the sign of f'(x).

Concave up

Graph bends like a cup; tangent lines tend to lie below the curve; slopes increase as x increases.

Concave down

Graph bends like a cap; tangent lines tend to lie above the curve; slopes decrease as x increases.

Concavity via second derivative

If f''(x)>0 on an interval, f is concave up there; if f''(x)<0, f is concave down there.

Inflection point

A point on the graph where concavity changes (concave up to concave down or vice versa).

Second Derivative Test

For a critical point where f'(c)=0: if f''(c)>0 it’s a local min; if f''(c)<0 it’s a local max; if f''(c)=0, inconclusive.

Objective function (optimization)

The quantity you want to maximize or minimize (e.g., area, cost, surface area), written as a function to optimize.

Constraint (optimization)

A condition that links variables (e.g., fixed perimeter or fixed volume) used to rewrite the objective function in one variable and determine a feasible domain.