AP Calculus AB - Chapter 3

Definition of Extrema

Let f be defined on an interval I containing c.

1. f(c) is the minimum of f on I when f(c) is </= f(x) for all x in I.

2. f(c) is the maximum of f on I when f(c) is >/= f(x) for all x in I.

Also called absolute minimum and absolute maximum

Extreme Value Theorem

If f is continuous on a closed interval [a, b], then f has both a minimum and a maximum on the interval.

Definition of Relative Extrema

1. If there is an open interval containing c on which f(c) is a maximum, then f(c) is relative maximum of f

2. If there is an open interval containing c on which f(c) is a minimum, then f(c) is called a relative minimum of f

Also called local maximum and local minimum

Definition of a Critical Number

Let f be defined at c. If f’(c) = 0 or if f is not differentiable at c, then c is a critical number of f.

Finding Extrema on a Closed Interval

1. Determine if interval is given (if given, it’s absolute)

2. Find critical numbers using derivative, solve for x

3. Find function value using original function

4. Determine maxima/minima

Remember

• Critical numbers = x-values

• Maximas/minimas = y-values

Rolle’s Theorem

Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If f(a) = f(b), then there is at least one number in c in (a, b) such that f’© = 0.

Mean Value Theorem

Same thing as Rolle’s Theorem plus:

There exists a number c in (a, b) such that f’© = [f(b)-f(a)]/b-a

The slope of the tangent line is equal to the slope of the secant line drawn between two endpoints.

Finding x-intercepts and showing f’© = 0 at some point between x-intercepts

1. Determine x-intercepts

2. Find derivative, solve for x

3. Is the x from the derivative in between the intercepts?

Finding out whether or not the Mean Value Theorem applies, finding c

1. Is the function continuous and differentiable?

2. If so, find the derivative of the function

3. Find the average rate of change using given interval (secant line)

4. Set the derivative equal to the average rate of change, solve for x, determine values of c

Increasing and Decreasing Functions

Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b),

1. If f’(x) > 0 for all x in (a, b), then f is increasing on [a, b]

2. If f’(x) < 0 for all x in (a, b), then f is decreasing on [a, b]

3. If f’(x) = 0 for all x in (a, b), then f is constant on [a, b]

Steps for Finding Increasing/Decreasing/Constant

1. Find critical numbers to determine extremas

2. Draw rough sign chart and substitute a value from each interval into f’(x) to test it

   1. f’(x) on top showing positive or negative slopes

   2. f(x) on bottom showing increasing or decreasing slopes

3. Indicate how the function behaves from the rules above. Where does it increase? Decrease? Constant?

The First Derivative Test

Let c be a critical number of a function f that is continuous on an open interval I containing c. If f is differentiable on the interval, except possibily at c, then f© can be classified as follows:

1. If f’(x) changes from - → + at c, then f has a minimum at (c, f©)

2. If f’(x) changes from + → - at c, then f has a maximum at (c, f©)

3. If f’(x) is + on both sides of c or - on both sides of c, then f© is neither a relative maximum nor a relative minimum

Using First Derivative Test to find extremas

After finding critical points and determining where function increases/decreases/constant, plug critical numbers and given interval into original function to determine extremas. Are the critical numbers where the minima and maxima are?

Concavity

Let f be differentiable on open interval I. The graph of f is concave upward on I if f’ is increasing on the interval and concave downward on I if f’ is decreasing on the interval.

Test for Concavity

Let f be a function whose second derivative exists on open interval I

1. If f’’(x) > 0 for all x in I, then the graph of f is concave upward on I.

2. If f’’(x) < 0 for all x in I, then the graph of f is concave downward on I.

Relationship between original function, first derivative, and second derivative

• If the second function is positive, then the first derivative is increasing and the original function is concave up.

• If the second derivative is negative, then the first derivative is decreasing and the original function is concave down.

Guidelines for Determining Concavity

*Similar process to First Derivative Test

1. Find the second derivative of the function.

2. Locate the x-values where f’’(x) = 0 and f’’(x) is undefined

3. Make a sign chart for the second derivative using the x-values you found in part 2.

4. In areas where the second derivative is positive, f(x) is concave up. In areas where f’’(x) is negative, f(x) is concave down.

Points of Inflection

Let f be a function that is continuous on an open interval and let c be a point in the interval. If the graph of f has a tangent line at this point (c, f©), then this point is a point of inflection of the graph of f if the concavity of f changes from upward to downward or downward to upward at the point.

Second Derivative Test

Let f be a function such that f’© = 0 and the second derivative of f exists on an open interval containing c.

1. If f’’© > 0, then f has a relative minimum at (c, f©). Concave up.

2. If f’’© < 0, then f has a relative maximum at (c, f©). Concave down.

If f’’© = 0, then the test fails. That is, f may have a relative minimum, relative maximum, or neither. In such cases, you can use the first derivative test.

Definition of Linearization

If f is differentiable at x=a, then the equation of the tangent line,

L(x) = f(a) + f’(a)(x-a)

defines the linearization of f at a. The approximation f(x) approximately equal to L(x) is the standard linear approximation of f and a. The point x = a is the center of the approximation.

How to find a linearization (linear approximation) of f(x) centered at x = c to approximate f(x) at x = a, a value near the center x = c

1. Find the equation of the tangent line at the center (c, f©) in point-slope form.

2. Solve for y and rename it L(x).

3. Plug in x = a into L(x) writing the notation VERY CAREFULLY as f(a) is approximately equal to L(a) = …

4. If asked, determine if L(a) is an overestimate or underestimate by examining the concavity of f(x) at the center x = c

   1. If f’’© < 0, f(x) is concave down at x = c and L(a) is an overestimate.

   2. If f’’© > 0, f(x) is concave up at x = c and L(a) is an underestimate.

Rectilinear Motion

Three closely related concepts:

• Position: x(t) = s(t) *either one can be written

  • equivalent to f(x)

• Velocity: v(t) = x’(t)

  • equivalent to f’(x)

• Acceleration: a(t) = v’(t) = x’’(t)

  • equivalent to f’’(x)

If x(t) represents the position of a particle along the x-axis at any time t, then the following statements are true:

1. “Initially” means when t=0

2. “At the origin” means x=0

3. “At rest” means v=0

4. If the velocity of the particle is positive, then the particle is moving to the right.

5. If the velocity of the particle is negative, then the particle is moving to the left.

6. To find the average velocity over a time interval, divide the change in position by the change in time.

   1. Secant line: [f(b)-f(a)]/(b-a)

7. Instantaneous velocity is the velocity at a single moment (instant!) in time.

8. If the acceleration of the particle is positive, then the velocity is increasing.

9. If the acceleration of the particle is negative, then the velocity is decreasing.

10. In order for a particle to change direction, the velocity must change signs.

    1. Positive to negative

    2. Negative to positive

11. One way to determine total distance traveled over a time interval is to find the sum of the absolute values of the differences in position between all resting points.

Steps to approaching optimization problems

1. What is being maximized or minimized? Determine through words and an equation.

2. What are the constraints? Do you need to draw a picture? Determine through words and an equation.

3. Solve the constraint equation for one variable.

4. Use the constraint equation to rewrite the max/min function in terms of one variable and simplify it.

5. Find the critical points. Determine the absolute max or min.

6. Read the problem again. Have you answered it? Does your answer make sense in the problem? Write a sentence to answer the question.