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

Calculus 3.1: Derivatives of Inverse Functions

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)

D. Derivatives of Inverse Trig Functions

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 line @ (0,1) with slope f ‘(0)=1> Ex 1: fig 10,11 p.427 (Stewart)

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:

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

B. General Log Functions: Using Chain Rule:

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

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>

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]

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

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

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

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:

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

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

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:

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;

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!