Derivative Applications

Question 1

Increasing and Decreasing

• If f’(x) > 0 on an interval, then f is _ on that interval.
• If f’(x) < 0 on an interval, then f is _ on that interval.
• Split up intervals at asymptotes.

Answer 1

Increasing, decreasing

Question 2

Largest maximum

Answer 2

Absolute maximum

Question 3

Smallest minimum

Answer 3

Absolute minimum

Question 4

Any maximum

Answer 4

Local maximum (relative)

Question 5

Any minimum

Answer 5

Local minimum (relative)

Question 6

A number c in the domain of f such that either f’(c) = 0 or f’(c) does not exist.

Answer 6

Critical numbers

Question 7

(x, y)

Answer 7

Point

Question 8

f(x)

Answer 8

Value

Question 9

Local Maximum and Minimum

• Suppose f’’ is continuous near c.
• If f’(c) = 0 and f’’(c) > 0, then f has a local _ at c.
• If f’(c) = 0 and f’’(c) < 0, then f has a local _ at c.
• If f has a local maximum or minimum at c, then c is a _ _ of f.

Answer 9

Minimum, maximum, critical number

Question 10

Absolute Maximum and Minimum

• Find the values of f at critical numbers of f in (a, b).
• Find the values of f at the _ of the interval.
• The largest value is the absolute _ value and the lowest is the absolute _ value.

Answer 10

Endpoints, maximum, minimum

Question 11

Minimum, Maximum, and Inflection Point

• Suppose that c is a critical number of a continuous function f.
• If f’ changes from positive to negative at c, then f has a local _ at c.
• If f’ changes from negative to positive at c, then f has a local _ at c.
• If f’ is positive to the left and right of c, or negative to the left and right of c, then f has an _ _ at c.

Answer 11

Maximum, minimum, inflection point

Question 12

Concavity

• If f’’(x) > 0 for all x in I, then the graph of f is concave _ on I.
• If f’’(x) < 0 for all x in I, then the graph of f is concave _ on I.
• If f’’(x) = 0 or dne, then _.

Answer 12

Upward, downward, inflection point

Question 13

Asymptotes
-Always write as a line
-Vertical Asymptote: Set denominator equal to _
Horizontal Asymptote: x/x means _, 1/x means _, x/1 means _ _
Slant Asymptotes: Denominator | numerator and omit _

Answer 13

Zero, coefficient/coefficient, 0, slant asymptote, remainder

Question 14

Curve Sketch
Domain: Set of x values (exclude asymptotes)
Intercepts: Set _ and _
Symmetry: If _ then even function. If _ then odd function. If _ then periodic function.
Asymptotes
Increase or decrease
Local and Absolute Maximums and Minimums
Concavity and Points of Inflection
Sketch Curve: Draw asymptotes, plot points, line following criteria.

Answer 14

y = 0, x = 0, f(-x) = f(x), f(-x) = -f(x), f(x+p) = f(x)

Question 15

Mean Value Theorem
-Let f be a function that satisfies the following hypotheses:
-f is _ on the closed interval [a, b].
-f is _ on the open interval (a, b).
-Then there is a number c in (a, b) such that
f’(c) = [f(b) - f(a)] / (b-a) or f(b) - f(a) = f’(c)(b - a)

Answer 15

Continuous, differentiable

Question 16

Rolle’s Theorem
Let f be a function that satisfies the following three hypotheses:
1. f is _ on the closed interval [a, b].
2. f is _ on the open interval (a, b).
3. f(a) = _
Then there is a number c in (a, b) such that f’(c) = _.

Answer 16

Continuous, differentiable, f(b), 0

Question 17

Anti-curve Sketch
Pay attention to range.
Pay attention to function (derivative ect.)
Pay attention to when < or <= and [ or ( to determine whether end points should be _.
Make sure to _ ranges every time y = 0.
Make sure justification always has identification.

Answer 17

Included, split