Graph Analysis for Calculus

Unit 5

Overview of Topic

We will continue to explore applications of the derivative. We will use the derivative to find optimal values of functions and to investigate properties of the graph of a function. This will include Mean Value Theorem, Extreme Value Theorem, 1st derivative test, 2nd derivative test, Candidates Test, Critical Numbers, Intervals of Increase and Decrease, and Concavity.

Theorems, Corollaries, and Definitions

Rolle's Theorem

Suppose f is a function defined on a closed interval [a, b].

If

f is continuous on [a, b]
f is differentiable on (a, b)
f(a) = f(b)

then there is at lest one number c in the open interval (a, b) for which f'(c) = 0.

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Mean Value Theorem (MVT)

Suppose f is a function defined on a closed interval [a, b].

If

f is continuous on [a, b]

f is differentiable on (a, b)

Then there is at least one number c in the open interval (a, b) for which

f'(c) = the slope on [a, b].

Corollary 1

If a function f is continuous on the closed interval [a, b] and is differentiable on the open interval (a, b) and if f'(x) = 0 for all numbers x in (a, b), the f is constant on (a, b).

Corollary 2

If the functions f and g are differentiable on an open interval (a, b) and if f'(x) = g'(x) for all numbers x in (a, b), then there is a number c for which f(x) = g(x) +c on (a, b).

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Corollary 3

Suppose f is a function that is differentiable on the open interval (a, b).

If f'(x) > 0 on (a, b), then f is increasing on (a, b)
If f'(x) < 0 on (a, b), then f is decreasing on (a, b)
In using the Increasing/Decreasing Function Test, we determine open intervals on which the derivative is positive or negative
The Increasing/Decreasing Function Test is valid if the interval (a, b)

(-♾️, b)
(a, ♾️)
(-♾️, ♾️)

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Definition of Maximum and Minimum

Absolute maximum and/or minimum points happen in three different cases

If f(x) is on a closed interval

If the end behavior on f(x) goes to negative infinity as x goes to positive infinity and negative infinity, then there could be a absolute maximum, but not an absolute minimum point

If the end behavior on f(x) goes to positive infinity as x goes to positive infinity and negative infinity, then there could be a absolute minimum, but not an absolute maximum point

Absolute minimum and maximum points will not share a value with anything else. If they do, then they are considered relative minimum or relative maximum points.

Extreme Value Theorem (EVT)

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

The location of the largest and smallest values happen at either of the following:

The value of f at the critical numbers in the open interval (a, b)
f(a) and f(b), the values of f at the endpoint a and b

Condition for a Local Maximum or Minimum

If a function f has a local maximum or a local minimum at the number c, then either f'(c) = 0 or f'(c) does not exist.

Fermat's Theorem

If a differentiable function f has a local maximum or minimum at c, then f'(c) = 0.

Candidates Test

1. Locate all critical numbers on the open interval (a, b)

2. Evaluate f at each critical number and at the endpoints a and b

3. The largest value is the absolute maximum and the smallest value is the absolute minimum

Definition of Critical Numbers

A critical number of a function f is a number c in the domain of f for which either f'(c) = 0 or f'(c) does not exist.

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First Derivative Test

Suppose f is a function that is continuous on an interval I. Suppose that c is a critical number of f and (a, b) is an open interval in I containing c.

If f'(x) > 0 for a < x < c and f'(x) < 0 for c < x < b, then f(c) is a local maximum value
If f'(x) < 0 for a < x < c and f'(x) > 0 for c < x < b, then f(c) is a local minimum value
If f'(x) has the same sign on both sides of c, then f(c) is neither a local maximum value not a local minimum value

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Test for Concavity

Suppose f is a function that is continuous on a closed interval [a, b] and f' and f" exist on the open interval (a, b).

If f"(x) > 0 on the interval (a, b), then f is concave up on (a, b)

If f"(x) < 0 on the interval (a, b), then f is concave down on (a, b)

If f"(x) = 0 on the interval (a, b), then f is changing concavity on (a, b)

Point of Inflection (POI)

Suppose f is a function that has a tangent line at every number in an open interval (a, b) contain c. If the tangent lines of f lie above the graph of f on one side of the point (c, f(c)) and the tangent lines of f lie below the graph on the other side of the point (c, f(c)) is an inflection point of f. This is where the graph of f changes concavity.

An inflection point happens with either f"(c) = 0 or if f"(c) does not exist.

To find POIs, do the following:

1. Find all numbers in the domain of f at which f"(x) = 0 or does not exists.

2. Test for concavity.

3. If concavity changes, then there is an inflection point at that c value, otherwise, there is not inflection point.

2nd Derivative Test

Suppose f is a function for which f' and f" exist on an open interval (a, b) and c lies in (a, b) and is a critical number of f.

If f"(c) < 0, then f(c) is a local maximum value

If f"(c) > 0, then f(c) is a local minimum value

Note:

If the second derivative of the unction does not exist at a critical number, the 2nd derivative test cannot be used.

If the second derivative exists at a critical number, but equals 0, the 2nd derivative test gives no information.

In both of these situations, the 1st derivative test must be used to identify local extreme points.

Particle Motion

Motion of an Object Moving on a Line

If the velocity (v) or the first derivative is greater than 0, the position x of the object is increasing with t, and the object is moving to the right.