AB Calculus - Hardtke

Notes 4.3: Deriv. Tests Name ________________________________

1. Rolle's Thrm: If f is differentiable over [a, b] and

if f(___) = f(___), then f'(c ) = ____ for at least one number c in _________.

2. Mean Value Theorem (MVT): If f is differentiable on [a, b], then there exists at least one c in (a, b)

such that f'(c ) = ________________ or equivalently, f(b) - f(a) = f'(c )(b - a).

3. Increasing/Decreasing Test:

i. If f'(x) > 0 for every x in (a, b), then f is _______________ (increasing or decreasing) on [a, b].

ii. If f'(x) < 0 for every x in (a, b), then f is _______________ (increasing or decreasing) on [a, b

4. First Derivative Test: Let c be a critical number of a continuous function f.

i. If f' changes from positive to negative at c, then f(c) is a ________________________ of f.

ii. If f' changes from negative to positive at c, then f(c) is a ________________________ of f.

iii. If f' does not change signs at c, then f(c) is not a _____________________ of f.

5. Definition of Concavity: The graph of f is

i. Concave _______________ (Upward or Downward) if f' is increasing on I (i.e., interval I)

ii. Concave _______________ (Upward or Downward) if f' is decreasing on I

6. Test for Concavity: The graph of f is

i. Concave _______________ (Upward or Downward) if f"(x) > 0 on I

ii. Concave _______________ (Upward or Downward) if f"(x) < 0 on I

7. Definition of Pt of Inflection: A point (c, f(c) ) on a graph of f is a point of inflection if:

i. f is continuous at c AND

ii. There is an open interval (a, b) containing c such that the graph is concave upward on (a, c) and

concave _______________ on (____, _____), or vice versa.

*You can shorten this by saying "the concavity changes at P(c, f(c) )."

8. Second Derivative Test: Suppose that f is continuous near c.

i. If f ’(c) = 0 and f"(c) < 0, then f has a local ___________ at c.

ii. If f ’(c) = 0 and f"(c) > 0, then f has a local ___________ at c.

Warning: If f"(c) = 0, the second derivative test is not applicable. In such cases, use first deriv. test.

9. Closed Interval Test: When a problem involves a closed interval, don't forget that global extrema can

occur at _________ _____ or at either _____________, so you must compare these ____-coordinates.

10. IVT: Given f is cont. over [a, b], for any N such that f(a) ≤ N ≤ f(b), there exists a c in (a,b) such that f(c) = N.

11. Extreme Value Theorem: Given f is cont. over [a, b], then f must have an absolute max & absolute min in [a, b].

AB Calc Assignment Page

Assignments & Opportunities:

I will TRY to have Sketchpad projects back to you next Monday or Tuesday.

Tomorrow: p268; 5,22,27,45 & p280; 9

Do warm-up immediately w/ your partner

In large group, we will complete the Notes & Sample Problem handout for Sections 4.2 & 4.3 -

Extrema and Derivative Tests

Study and be able to use AND identify each idea and theorem from these notes.

Today's Topics & Class Plan:

Warm-up: Warming Up to Derivatives

f ‘ (3) = 2, f”(3) = 4 2. f ‘ (3) = 2, f”(3) = – 41.

Let f(t) be the temperature at time t where you live and suppose that at time t = 3 you feel

uncomfortably hot. How do you feel about the given data in each case below?

f ‘ (3) = – 2, f”(3) = 4 4. f ‘ (3) = – 2, f”(3) = – 43.

.

AB Calc Sect 4.3 - NotesMonday, November 28, 2011

Chapter 4 Page 1

http://faculty.muhs.edu/hardtke/ABCalc_Assignments.htm

AB Calculus - Hardtke

Notes 4.3: Deriv. Tests Name ________________________________

1. Rolle's Thrm: If f is differentiable over [a, b] and

if f(___) = f(___), then f'(c ) = ____ for at least one number c in _________.

2. Mean Value Theorem (MVT): If f is differentiable on [a, b], then there exists at least one c in (a, b)

such that f'(c ) = ________________ or equivalently, f(b) - f(a) = f'(c )(b - a).

3. Increasing/Decreasing Test:

i. If f'(x) > 0 for every x in (a, b), then f is _______________ (increasing or decreasing) on [a, b].

ii. If f'(x) < 0 for every x in (a, b), then f is _______________ (increasing or decreasing) on [a, b

4. First Derivative Test: Let c be a critical number of a continuous function f.

i. If f' changes from positive to negative at c, then f(c) is a ________________________ of f.

ii. If f' changes from negative to positive at c, then f(c) is a ________________________ of f.

iii. If f' does not change signs at c, then f(c) is not a _____________________ of f.

5. Definition of Concavity: The graph of f is

i. Concave _______________ (Upward or Downward) if f' is increasing on I (i.e., interval I)

ii. Concave _______________ (Upward or Downward) if f' is decreasing on I

6. Test for Concavity: The graph of f is

i. Concave _______________ (Upward or Downward) if f"(x) > 0 on I

ii. Concave _______________ (Upward or Downward) if f"(x) < 0 on I

7. Definition of Pt of Inflection: A point (c, f(c) ) on a graph of f is a point of inflection if:

i. f is continuous at c AND

ii. There is an open interval (a, b) containing c such that the graph is concave upward on (a, c) and

concave _______________ on (____, _____), or vice versa.

*You can shorten this by saying "the concavity changes at P(c, f(c) )."

8. Second Derivative Test: Suppose that f is continuous near c.

i. If f ’(c) = 0 and f"(c) < 0, then f has a local ___________ at c.

ii. If f ’(c) = 0 and f"(c) > 0, then f has a local ___________ at c.

Warning: If f"(c) = 0, the second derivative test is not applicable. In such cases, use first deriv. test.

9. Closed Interval Test: When a problem involves a closed interval, don't forget that global extrema can

occur at _________ _____ or at either _____________, so you must compare these ____-coordinates.

10. IVT: Given f is cont. over [a, b], for any N such that f(a) ≤ N ≤ f(b), there exists a c in (a,b) such that f(c) = N.

11. Extreme Value Theorem: Given f is cont. over [a, b], then f must have an absolute max & absolute min in [a, b].

Chapter 4 Page 2

12. In which graph below are the slopes of the tangents increasing? Does this coordinate with

CU or CD?.

13. Visualize the Second Derivative Test using graphs A and B above. If f ‘ (c) = 0 in a

CD interval, could (c, f(c)) be a local min?

14. Use the second deriv. test to find local extrema of f(x) = 12 + 2x2 - x4, inflection

pts and concavity.

Step 1: f'(x) = Thus, crit #s are:

Step 2: f"(x) = Now evaluate f" at the crit numbers:

local maxima: local minima:

Step 3: Use signs of f" & plug into f to determine the appropriate y-coordinates for

Step 4: To locate pts of inflection & concavity, solve f" for zero and examine sign of

f"(x) in each interval:

Chapter 4 Page 3

Use the First Derivative Test to find the extrema of f(x) = x3 – 6x2 + 9x + 2 15.

over [–∞,∞].

Step 1: f'(x) = Thus, crit #s are:

Step 2: Check where f ‘ changes signs on each interval determined by the crit #s.

Find the extrema of f(x) = x3 – 6x2 + 9x + 2 over [–1, 4]. (Same function as in question 15)16.

.

Chapter 4 Page 4

Chapter 4 Page 5

Chapter 4 Page 6

Chapter 4 Page 7