Calculus So Far Weve learned about what a

Calculus So Far… We’ve learned about: -what a limit means -how to calculate limits of all kinds of functions -what can happen to a function (continuity) -how to use limits to find a derivative -what the derivative means -calculating the derivative of all kinds of functions Left for this semester: -applications of the derivative Next semester: -the “antiderivative” or integral

Relative Extrema 5. 1

Definition of Extrema � Let f be defined on an interval I containing c. 1. f (c) is the minimum of f on I if for all x in I. 2. f (c) is the maximum of f on I if for all x in I.

The Extreme Value Theorem �If f is continuous on a closed interval [a, b], then f has both a minimum and a maximum on the interval.

Definition of Relative Extrema If there is an open interval containing c on which f (c) is a maximum, the f (c) is called a relative maximum of f, or you can say that f has a relative maximum at (c, f (c)). 2. If there is an open interval containing c on which f (c) is a minimum, the f (c) is called a relative minimum of f, or you can say that f has a relative minimum at (c, f (c)). 1.

Find the value of the derivative at each of the relative extrema shown below. (π/2, 1) (3π/2, -1)

Critical Numbers �Let f be defined at c. If or if f is not differentiable at c, then c is a critical number of f. �If f has a relative minimum or relative maximum at x = c, then c is a critical number of f.

Guidelines for Finding Extrema on a Closed Interval � To find the extrema of a continuous function f on a closed interval [a, b], use the following steps. 1. 2. 3. 4. Find the critical numbers of f in (a, b). Evaluate f at each critical number in (a, b). Evaluate f at each endpoint of [a, b]. The least of these values is the minimum. The greatest is the maximum.

Example �Find the extrema of the interval [ 1, 2]. on

Homework �p. 198 ~ 1 -10, 11 -29 (O) , 35 -41 (O)

Reflection 1. Rate the level of difficulty of the mock AP Test from 1 -10. 2. What the most difficult part of the exam? Why? 3. What was the easiest part of the exam? Why?

Rolle’s Theorem and the Mean Value Theorem 5. 2

Find the extrema of the function on the given interval

Rolle’s Theorem �Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If then there is at least one number c in (a, b) such that.

Rolle’s Theorem �Find the two x-intercepts of and show that at some point between the two x-intercepts. �Find that point.

Rolle’s Theorem �Let . �Find all values of c in the interval ( 2, 2) such that.

The Mean Value Theorem �If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number c in (a, b) such that

Mean Value Theorem �Given , find all values of c in the open interval (1, 4) such that

Mean Value Theorem �Given of c in the open interval that , find all values such

Mean Value Theorem – Florida Freeway Tolls � Two stationary patrol cars equipped with radar are 5 miles apart on a highway. As a truck passes the first patrol car, its speed is clocked at 55 miles per hour. Four minutes later, when the truck passes the second patrol car, its speed is clocked at 50 miles per hour. Prove that the truck must have exceeded the speed limit of 55 miles per hour at some time during the 4 minutes.

Homework �p. 206/ 1 -7 odd, 9 -14, 51 -56 �Rolle’s Theorem Worksheet

Increasing and Decreasing Functions and the First Derivative Test 5. 3

Definition of Increasing and Decreasing Functions �A function f is increasing on an interval if for any two numbers x 1 and x 2 in the interval, implies. �A function f is decreasing on an interval if for any two numbers x 1 and x 2 in the interval, implies

Test for Increasing and Decreasing Functions � Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 1. If for all x in (a, b), then f is increasing on [a, b]. 2. If for all x in (a, b), then f is decreasing on [a, b]. 3. If for all x in (a, b), then f is constant on [a, b].

Intervals of an increasing and decreasing function �Find the open intervals on which is increasing and decreasing.

a)Find the critical numbers of f (if any), b)Find the open interval(s) on which the function is increasing or decreasing.

Homework �p. 202/(15 -27 bc)odd, 39 -42

The First Derivative Test � Let c be a critical number of a function f that is continuous on an open interval I containing c. If f is differentiable on the interval, except possibly at c, then f (c) can be classified as follows. 1. If changes from negative to positive at c, then f has a relative minimum at. 2. If changes from positive to negative at c, then f has a relative maximum at. 3. If is positive on both sides of c or negative on both sides of c, then f is neither a relative minimum nor a relative maximum.

First Derivative Test �Find �in the relative extrema of the function the interval .

First Derivative Test �Find the relative extrema of the function

Homework �p. 215/1 -5 odd, 21 a, 22 a, 23 - 25, 27, 45

Concavity and the Second Derivative Test 4. 3

Concavity �Let f be differentiable on an open interval I. The graph of f is concave upward on I if f ’ is increasing on the interval and concave downward on I if f ‘ is decreasing on the interval.

Test for Concavity Let f be a function whose second derivative exists on an open interval I. 1. If for all x in I, then the graph of f is concave upward on I. 2. If for all x in I, then the graph of f is concave downward on I. �

Concavity �Determine the open intervals on which the graph of �is concave upward or downward.

Definition of Point of Inflection �Let f be a function that is continuous on an open interval and let c be a point in the interval. If the graph of f has a tangent line at this point (c, f(c)), then this point is a point of inflection of the graph of f if the concavity of f changes from upward to downward (or downward to upward) at the point.

Points of Inflection �If (c, f(c)) is a point of inflection of the graph of f , then either or does not exist at x = c.

Concavity �Determine the open intervals on which the graph of �is concave upward or downward.

Homework �p. 215/ 7 -19 odd, 21 b, 22 b, 46 -48

The Second Derivative Test �Let f be a function such that and the second derivative of f exists on an open interval containing c. 1. If , then f has a relative minimum at. 2. If , then f has a relative maximum at. 3. If , the test fails. In that case, use the first derivative test.

Second Derivative Test �Find . the relative extrema for

Quick Curve Sketching

Sketching a graph �Sketch the graph of

Homework �p. 216/ 33 -39 odd, 49 -52

Complete Curve Sketching

Sketch the Complete Graph

Optimization 4. 4

Finding Two Numbers �Find two numbers whose sum is 20 and whose product is as large as possible.

Guidelines for Solving Applied Minimum and Maximum Problems �Identify all given quantities and quantities to be determined. If possible, make a sketch. �Write a primary equation for the quantity that is to be maximized or minimized. �Reduce the primary equation to one having a single independent variable. This may involve the use of secondary equations relating the independent variables of the primary equation. �Determine the feasible domain of the primary equation. That is, determine the values for which the stated problem makes sense. �Determine the desired maximum and minimum value by calculus techniques previously discussed.

Manufacturing �A manufacturer wants to design an open box having a square base and a surface area of 108 square inches. What dimensions will produce a box with maximum volume?

Homework �p. 226/ 1 -11 odd

Example � A farmer has 100 yards of fencing to form two identical rectangular pens and a third pen that is twice as long as the other two pens, as shown in the diagram. All three pens have the same width, x. Which value of y produces the maximum total fenced area? A. x B. 10 C. x D. E. none of these y y

Minimizing Distance �Which points on the graph of are closest to the point (0, 2)?

More Manufacturing �A cylindrical can is to be made to hold 1 L of oil. Find the dimensions that will minimize the cost of the metal to manufacture the can.

Homework �p. 226/ 13 -21 odd, 30

Cultivating an Orchard �An apple orchard has 24 trees on a one acre lot, each tree yielding 600 apples. The farmer knows from past experience that for each extra tree that is planted on the acre, 12 less apples grow on each tree. What is the optimum number of trees to yield the maximum number of apples?

Newton’s Method 4. 5

The Background y a x 1 c x b x

The Background y a x 1 x 3 c x 2 b x

Newton’s Method for Approximating the Zeros of a Function � Let f (c) = 0, where f is differentiable on an open interval containing c. Then, to approximate c, use the following steps. 1. Make an initial estimate x 1 that is close to c. 2. Determine a new approximation 3. If is within the desired accuracy, let xn+1 serve as the final approximation. Otherwise, return to Step 2 and calculate a new approximation.

Using Newton’s Method �Calculate three iterations of Newton’s Method to approximate a zero of. Use x 1 = 1 as the initial guess. 1 2 3 4

Using Newton’s Method �Use Newton’s Method to approximate the zeros of. Continue the iterations until two successive approximations differ by less than 0. 0001.

Points of Intersection

Homework �p. 243/ 11 -14

Linearization 4. 5

Linear Approximation �Find the tangent line approximation (Linearization) of at the point (0, 1). Then use the tangent line approximation to find the value of

Linearization �If f is differentiable at x = a, then the equation of the tangent line, defines the linearization of f at a. The approximation f (x) ≈ L(x) is the standard linear approximation of f at a. The point x = a is the center of the approximation.

Tangent Line Approximation �Find the tangent line approximation for near x = 0.

c c

Definition of Differentials �Let y = f (x) represent a function that is differentiable on an open interval containing x. The differential of x is any nonzero real number. The differential of y is

Comparing dy and �Let Find dy when x = 1 and dx = 0. 01. Compare this value with for x = 1 and.

Finding the Differential of a Function �Find the differential dy of the function

Homework �p. 242/ 1 -7 odd, 19 -29 odd

Propagated Error

Propagated Error �The radius of a ball bearing is measured to be 0. 7 inch. If the measurement is correct to within 0. 01 inch, estimate the propagated error in the volume of V of the ball bearing.

Error Propagation �The measurement of the base and altitude of a triangle are found to be 36 and 50 centimeters, respectively. The possible error in each measurement is 0. 25 centimeter. Use differentials to approximate the possible propagated error in computing the area of the triangle.

Error Propagation �The measurement of a side of a square is found to be 15 centimeters, with a possible error of 0. 05 centimeter. �Approximate the percent error in computing the area of the square. �Estimate the maximum allowable percent error in measuring the side if the error in computing the area cannot exceed 2. 5%

Homework �p. 243/ 31 -37 odd, 41 -51 odd

Approximating Values

Approximating Values �Use differentials to approximate

Homework �p. 242/ 8, 11 -14

Related Rates 4. 6

Related Rates �Suppose x and y are differentiable functions of t and are related by the equation �Find dy / dt when x = 1, given that dx / dt = 2 when x = 1.

Real-life problem �A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate of 1 foot per second. When the radius is 4 feet, at what rate is the total area A of the disturbed water changing?

Guidelines for Solving Related. Rate Problems 1. 2. 3. 4. Identify all given quantities and quantities to be determined. Make a sketch and label the quantities. Write an equation involving the variables whose rates of change either are given or are to be determined. Using the Chain Rule, implicitly differentiate both sides of the equation with respect to t. After completing Step 3, substitute into the resulting equation all known values for the variables and their rates of change. Then solve for the required rate of change.

Another Example �A baseball diamond is a square whose sides are 90 ft long. Suppose that a player running from second base to third base has a speed of 30 ft/s at the instant when he or she is 20 ft from third base. At what rate is the player’s distance from home plate changing at that instant?

Homework �p. 251/ 1 -5 odd, 9 -15 odd

Example �A swimming pool is 12 meters long, 6 meters wide, 1 meter deep at the shallow end and 3 meters deep at the deep end. Water is being pumped into the pool at ¼ cubic meter per minute, and there is 1 meter of water at the deep end. At what rate is the water level rising?