1. Calculus 3.1: Derivatives of Inverse Functions
A. Review: Inverse Functions:
B. The Derivative of an Inverse Function:

2. Calculus 3.1: Derivatives of Inverse Functions

3. Examples:
Given f(x) = x2 (x > 0), find (f -1)’ (1)
Given f(x) = 2x + cos x, find (f -1)’ (1)
Find the value of (f -1)’ at x = f(a)

4. D. Derivatives of Inverse Trig Functions

6. Calculus 3.2: Exponential Functions
A. Exponential Function:
B. Definition of the number e :
e is the number such that
<f(x)=ex has a tangent (0,1) with slope f ‘(0)=1>
Ex 1: fig 10,11 p.427 (Stewart)

7. C. Derivative of an Exponential Function:
1. General rule:
2. Derivative of the natural exponential function:
because ln e = 1
3. Using the Chain Rule:

8. Calculus 3.3: Derivatives of Logarithmic Functions
A. Natural Log Function:
Using Chain Rule:
or:

9. B. General Log Functions:
Using Chain Rule:

10. Calculus 3.4: Maximum/Minimum Values
A. Definitions
1. absolute maximum: f(c) > f(x) for all x in Domain
2. absolute minimum: f(c) < f(x) for all x in Domain
3. local (relative) maximum: f(c) > f(x) when x is “near” c
4. local (relative) minimum: f(c) < f(x) when x is “near” c

11. B. Extreme Value Theorem
If f is continuous on a closed interval [a,b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some #’s c and d in [a,b]
Fermat’s Theorem:
If f has a local maximum or minimum at c, and if f ‘(c) exists, then f ‘(c) = 0
<c is called a critical number of f if f ‘(c) = 0 or DNE>

12. D. Finding Absolute Max/Min Values on [a,b]
1. Determine the critical #(s) c
2. Find the values of f at the critical #(s) f(c)
3. Find the values of f at the endpoints of the interval f(a) and f(b)
4. Compare all values from Step 2 and 3
Ex 1: find critical #s:
Ex 2: find absolute max/min values of
f(x) = x3 – 3x2 + 1 on [-½,4]

13. Calculus 3.5: Mean Value Theorem
A. Rolle’s Theorem:
Let f be a function that satisfies the following hypotheses:
1. f is continuous on [a,b]
2. f is differentiable on (a,b)
3. f(a) = f(b)
Then there is a number c in (a,b) such that f ‘(c) = 0

14. B. Mean Value Theorem
Let f be a function that satisfies the following hypotheses:
1. f is continuous on [a,b]
2. f is differentiable on (a,b)
Then there is a number c in (a,b) such that

15. Calculus 3.6: Derivatives and Graphs of f(x)
A. Increasing/Decreasing Test
1. If f ‘(x) > 0 on an interval, then f is increasing on that interval
2. If f ‘(x) < 0 on an interval, then f is decreasing on that interval
Ex 1: find I/D intervals for f(x) = 3x4 – 4x3 – 12x2 + 5

16. B. First Derivative Test
(Suppose that c is a critical # of a continuous function f)
1. If f ‘ changes from positive to negative at c, then f has a local maximum at c
2. If f ‘ changes from negative to positive at c, then f has a local minimum at c
3. If f ‘ does not change sign at c, then f has no local minimum or maximum at c
Ex 2: a) find local max/min values for Ex 1
b) find local max/min values for g(x) = x + 2 sin x on
c) find local extrema for:

17. Calculus 3.7: Concavity & Points of Inflection
A. Concavity Test
1. If f”(x) > 0 for all x in I, then the graph of f is concave upward on I
<f lies above all its tangents on I>
2. If f”(x) < 0 for all x in I, then the graph of f is concave downward on I
<f lies below all its tangents on I>
see fig 4.21, 4.22 p.207-8

18. B. DEF: Point of Inflection:
A point P on y = f(x) where f is continuous and the curve changes from concave upward to concave downward or vice versa at P
See fig 4.23 p.208
C. Second Derivative Test
(suppose f” is continuous near c)
1. If f ‘(c) = 0 and f”(c) > 0, then f has a local minimum at c
2. If f ‘(c) = 0 and f”(c) < 0, then f has a local maximum at c
Ex 1: Find local extrema, concavity, and inflection pts:
a) y = x4 – 4x3

19. Calculus: Review for Unit Test 3
1. Find I/D intervals:
f(x) = x3 – 6x2 – 15x + 4
2. Find max/min points:
3. Find intervals of concavity:
4. Sketch the graph:

20. Given the graph of f ‘(x) above,
sketch f(x) if f(0) = 0
6. Sketch f(x) given the following: f is odd;
f ‘(x) < 0 for 0 < x < 2; f ‘(x) > 0 for x > 2;
f”(x) > 0 for 0 < x < 3; f”(x) < 0 for x > 3;

21. Calculus Unit 3 Test
Grademaster #1-25 (Name, Date, Subject, Period, Test Copy #)
Do Not Write on Test! Show All Work on Scratch Paper!
Label BONUS QUESTIONS Clearly on Notebook Paper. (If you have time)
Find Something QUIET To Do When Finished!