Calculus Date 121713 Obj SWBAT apply first derivative

• Calculus Date: 12/17/13 Obj: SWBAT apply first derivative test http: //youtu. be/PBKntt. VMb. V 4 first derivative test inc. dec. Today – Cover the first derivative test In class: Start WS 3 -3 A; complete for homework st) Tomorrow Cover the second derivative test (easier than 1 "Do not judge me by my successes, judge me by how many times Friday: With Christian complete any I fell down and got back up again. “ Nelson Mandela remaining worksheets, make sure You understand material from the test. , etc. • Announcements: • Break Packet online on Friday • Merry Christmas if I don’t see you

Increasing/Decreasing/Constant

Increasing/Decreasing/Constant

Increasing/Decreasing/Constant

The First Generic Example A similar Derivative Observation Test Applies at a Local Max.

The First Derivative Test Determine the sign of the derivative of f to the left and right of the critical point. left right conclusion f (c) is a relative maximum f (c) is a relative minimum No change No relative extremum

The First Derivative Test Find all the relative extrema of Relative max. f (0) = 1 Relative min. f (4) = -31 + 0 0 - 0 + 4 Evaluate the derivative at points on either side of extrema to determine the sign.

The First Derivative Test Sketch of function based on estimates from the first derivative test.

The First Derivative Test Here is the actual function.

Another Example Find all the relative extrema of Stationary points: Singular points:

Stationary points: Singular points: Evaluate the derivative at points on either side of extrema to determine the sign. Relative max. Use Yvars for Relative min. faster calculations ND – derivative not defined + ND + 0 -1 - ND 0 - 0 1 + ND +

Graph of Local max. + ND + 0 -1 - Local min. ND - 0 + ND + 0 1

Example: Graph There are roots at and . Possible extrema at Set . We can use a chart to organize our thoughts. First derivative test: y y=4 y=0 positive negative positive

Example: Graph There are roots at and . Possible extreme at First derivative test: Set y y=4 maximum at minimum at y=0 .

Example: Graph NOTE: On the AP Exam, it is not sufficient to simply draw the chart and write the answer. You must give a written explanation! First derivative test: There is a local maximum at (0, 4) because and for all x in (0, 2). There is a local minimum at (2, 0) because (0, 2) and for all x in

Example: Graph First derivative test: There is a local maximum at (0, 4) because and for all x in (0, 2). There is a local minimum at (2, 0) because (0, 2) and for all x in

Graph 1. Take the derivative f’(x) 2. Find the critical points f’(x) = 0; f’(x) = DNE 3. Make sign chart 1. Label critical points – put 0 or DNE on graph 2. Evaluate the derivative at points on either side of extrema to determine the sign. Use Yvars for faster calculations. Mark the chart as + or – in these areas 3. Evaluate f(x) at the critical points to determine actual values of extrema. 4. Add arrows to show increasing or decreasing regions in f(x) 4. Write out all extrema and use the value of the derivative and the appropriate interval to justify your answer.

Chapter 5 Applications of the Derivative Sections 5. 1, 5. 2, 5. 3, and 5. 4

Applications of the Derivative § Maxima and Minima § Applications of Maxima and Minima § The Second Derivative - Analyzing Graphs

Absolute Extrema Let f be a function defined on a domain D Absolute Maximum Absolute Minimum

Absolute Extrema A function f has an absolute (global) maximum at x = c if f (x) f (c) for all x in the domain D of f. The number f (c) is called the absolute maximum value of f in D Absolute Maximum

Absolute Extrema A function f has an absolute (global) minimum at x = c if f (c) f (x) for all x in the domain D of f. The number f (c) is called the absolute minimum value of f in D Absolute Minimum

Generic Example

Generic Example

Generic Example

Relative Extrema A function f has a relative (local) maximum at x c if there exists an open interval (r, s) containing c such that f (x) f (c) for all r x s. Relative Maxima

Relative Extrema A function f has a relative (local) minimum at x c if there exists an open interval (r, s) containing c such that f (c) f (x) for all r x s. Relative Minima

Generic Example The corresponding values of x are called Critical Points of f

Critical Points of f A critical number of a function f is a number c in the domain of f such that (stationary point) (singular point)

Candidates for Relative Extrema 1. Stationary points: any x such that x is in the domain of f and f ' (x) 0. 2. Singular points: any x such that x is in the domain of f and f ' (x) undefined 3. Remark: notice that not every critical number correspond to a local maximum or local minimum. We use “local extrema” to refer to either a max or a min.

Fermat’s Theorem If a function f has a local maximum or minimum at c, then c is a critical number of f Notice that theorem does not say that at every critical number the function has a local maximum or local minimum

Generic Example Two critical points of f that do not correspond to local extrema

Example Find all the critical numbers of Stationary points: Singular points:

Graph of Local max. Local min.

Extreme Value Theorem If a function f is continuous on a closed interval [a, b], then f attains an absolute maximum and absolute minimum on [a, b]. Each extremum occurs at a critical number or at an endpoint. a b Attains max. and min. a b Attains min. but no max. No min. and no max. Open Interval Not continuous

Example Find the absolute extrema of Critical values of f inside the interval (-1/2, 3) are x = 0, 2 Absolute Max. Absolute Min. Evaluate Absolute Max.

Example Find the absolute extrema of Critical values of f inside the interval (-1/2, 3) are x = 0, 2 Absolute Max. Absolute Min.

Example Find the absolute extrema of Critical values of f inside the interval (-1/2, 1) is x = 0 only Absolute Max. Evaluate Absolute Min.

Example Find the absolute extrema of Critical values of f inside the interval (-1/2, 1) is x = 0 only Absolute Max. Absolute Min.

Start Here a. Reviewing Rolle’s and MVT b. Remember n. Deriv and Yvars c. Increasing, Decreasing, Constant d. First Derivative Test

Finding absolute extrema on [a , b] 0. 1. 2. 3. 4. Verify function is continuous on the interval. Determine the function’s domain. Find all critical numbers for f (x) in (a , b). Evaluate f (x) for all critical numbers in (a , b). Evaluate f (x) for the endpoints a and b of the interval [a , b]. The largest value found in steps 2 and 3 is the absolute maximum for f on the interval [a , b], and the smallest value found is the absolute minimum for f on [a , b].

Rolle’s Theorem • Given f(x) on closed interval [a, b] – Differentiable on open interval (a, b) • If f(a) = f(b) … then – There exists at least one number a < c < b such that f ’(c) = 0 f(a) = f(b) a c b

The Mean Value Theorem (MVT) aka the ‘crooked’ Rolle’s Theorem We can “tilt” the picture of Rolle’s Theorem Stipulating that f(a) ≠ f(b) If f is continuous on [a, b] and differentiable on (a, b) There is at least one number c on (a, b) at which Conclusion: Slope of Secant Line Equals Slope of Tangent Line How is Rolle’s Connected to MVT? f(b) a c b f(a)

The Mean Value Theorem (MVT) aka the ‘crooked’ Rolle’s Theorem We can “tilt” the picture of Rolle’s Theorem Stipulating that f(a) ≠ f(b) If f is continuous on [a, b] and differentiable on (a, b) There is at least one number c on (a, b) at which Conclusion: The average rate of change equals the instantaneous rate of change f(b) evaluated at a point a c b f(a)

Finding c • Given a function f(x) = 2 x 3 – x 2 – Find all points on the interval [0, 2] where – Rolle’s? • Strategy – Find slope of line from f(0) to f(2) – Find f ‘(x) – Set f ‘(x) equal to slope … solve for x

CALCULATOR REQUIRED If A. 0 , how many numbers on [-2, 3] satisfy the conclusion of the Mean Value Theorem. B. 1 C. 2 D. 3 E. 4 f(3) = 39 f(-2) = 64 For how many value(s) of c is f ‘ (c ) = -5? X X X

Given the graph of f(x) below, use the graph of f to estimate the numbers on [0, 3. 5] which satisfy the conclusion of the Mean Value Theorem.

Relative Extrema Example: Find all the relative extrema of Stationary points: Singular points: None

First Derivative Test • What if they are positive on both sides of the point in question? • This is called an inflection point

Domain Not a Closed Interval Example: Find the absolute extrema of Notice that the interval is not closed. Look graphically: Absolute Max. (3, 1)

Optimization Problems 1. Identify the unknown(s). Draw and label a diagram as needed. 2. Identify the objective function. The quantity to be minimized or maximized. 3. Identify the constraints. 4. State the optimization problem. 5. Eliminate extra variables. 6. Find the absolute maximum (minimum) of the objective function.

Optimization - Examples An open box is formed by cutting identical squares from each corner of a 4 in. by 4 in. sheet of paper. Find the dimensions of the box that will yield the maximum volume. x 4 – 2 x x

Critical points: The dimensions are 8/3 in. by 2/3 in. giving a maximum box volume of V 4. 74 in 3.

Optimization - Examples An metal can with volume 60 in 3 is to be constructed in the shape of a right circular cylinder. If the cost of the material for the side is $0. 05/in. 2 and the cost of the material for the top and bottom is $0. 03/in. 2 Find the dimensions of the can that will minimize the cost top and bottom side

So Sub. in for h

Graph of cost function to verify absolute minimum: 2. 5 So with a radius ≈ 2. 52 in. and height ≈ 3. 02 in. the cost is minimized at ≈ $3. 58.

Second Derivative

Second Derivative - Example

Second Derivative

Second Derivative

Concavity Let f be a differentiable function on (a, b). 1. f is concave upward on (a, b) if f ' is increasing on aa(a, b). That is f ''(x) 0 for each value of x in (a, b). 2. f is concave downward on (a, b) if f ' is decreasing on (a, b). That is f ''(x) 0 for each value of x in (a, b). concave upward concave downward

Inflection Point A point on the graph of f at which f is continuous and concavity changes is called an inflection point. To search for inflection points, find any point, c in the domain where f ''(x) 0 or f ''(x) is undefined. If f '' changes sign from the left to the right of c, then (c, f (c)) is an inflection point of f.

Example: Inflection Points Find all inflection points of

Inflection point at x 2 - 0 2 +

The Point of Diminishing Returns If the function represents the total sales of a particular object, t months after being introduced, find the point of diminishing returns. S concave up on S concave down on The point of diminishing returns is at 20 months (the rate at which units are sold starts to drop).

The Point of Diminishing Returns S concave down on Inflection point S concave up on