Section Maximum and Minimum Values
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1 Section Maximum and Minimum Values Definition The number f(c) is a local maximum value of f if when x is near c. local minimum value of f if when x is near c. Example 1: For what values of x does f(x) have a local extrema? f(x) a b c d e x Definition: A critical number of a function f is a number c in the domain of f such that either f (c)=0 or f (c) does not exist. Example 2: Find the critical numbers of each of the given functions. a) f(x)=x 5 + 5x 4 75x b) g(x)=x 2/5 x 3/5 1
2 Definition Let c be a number in the domain D of a function f. Then f(c) is the absolute maximum value of f on D if for all x in D. absolute minimum value of f on D if for all x in D. Example 3: Find the absolute maximum and absolute minimum of each function if they exist. a) b) c) Example 4: Find the absolute max and min for the functions in Example 1 on the interval[ 1,1] The Closed Interval Method To find the absolute maximum and minimum values of a continuous function f on a closed interval[a,b]: 1. Find the values of f at the critical numbers of f in (a,b). 2. Find the values of f at the endpoints of the interval. 3. The largest of the values from Steps 1 and 2 is the absolute maximum values; the smallest of these values is the absolute minimum value. 2
3 Example 5: Find the absolute extrema of the following functions on the given intervals a) f(x)=2x 3 3x 2 12x+24 on[1,4] b) g(x)=x 3 + 3x 2 9x 7 on[ 2,0] c) h(x)=xlnx on [0.1,5] Section 4.2 Highly Suggested Homework Problems: 1, 3, 7, 9, 13, 21, 25, 29, 41, 45, 47, 49, 51 3
4 Increasing/Decreasing Test: Section Derivatives and the Shapes of Curves If f (x)>0 on an interval, then f is If f (x)<0 on an interval, then f is on that interval. on that interval. The First Derivative Test Suppose that x=c is a critical number of a continuous function f. 1. If f (x) changes from to at x=c, then we have that f(x) is and at x=c there is a. 2. If f (x) changes from to at x=c, then we have that f(x) is and at x=c there is a. 3. If the sign of f (x) is the same on both sides of x=c, then at x=c. Example 1: Determine the intervals where the following functions are increasing and decreasing and find the local extrema. a) f(x)=(x+2)e x b) g(x)=x 4 + 2x c) h(x)= x+3 5 x 4
5 Second Derivative Test for Local Extrema: Let c be a critical value for f(x). f (c) f (c) Concavity? f(c) is Example 2: Find the local extrema of f(x)= x x 5 using the Second Derivative Test for Local Extrema. Test for Concavity: For a function whose second derivative exists on an open interval(a,b): 1. If f (x)>0 for all x on (a,b), then f(x) is on (a,b). 2. If f (x)<0 for all x on (a,b), then f(x) is on (a,b). Definition: An inflection point is a point on the graph of the function where the. Example 3: Determine the intervals where the functions below are concave up and concave down and locate any inflection points. a) f(x)=x 4 6x 2 b) f(x)=ln(x 2 + 6x+13) 5
6 General Graphing Strategy: Step 1: Analyze f(x) Find the domain of f. Find asymptotes. Step 2: Analyze f (x): Create sign chart for f (x) and determine intervals of increasing/decreasing and local extrema. Step 3: Anlayze f (x): Create sign chart for f (x) and determine intervals of concave up, concave down, and inflection points. Step 4: Sketch the graph of f using all of the above information. Plot additional points as needed. Example 4: Use the graphing strategy to sketch a graph of each of the functions below: a) f(x)=3x 4 16x x
7 b) f(x)= 2x2 + 11x+14 x 2. 4 Section 4.3 Highly Suggested Homework Problems: 3, 5, 7, 11, 15, 17, 19, 21, 27, 29, 33, 37, 39, 41 7
8 Strategy for Solving Optimization Problems Section Optimization Problems 1. Introduce variables, look for relationships among these variables, and construct a mathematical model of the form Maximize (or minimize) f(x) on the interval I 2. Find the critical values of f(x). 3. Use the procedures developed in previous sections (and below) to find the absolute maximum (or minimum) value of f(x) on the interval I and the value(s) of x where this occurs. 4. Use the solution to answer all questions asked in the problem. Second Derivative Test for Absolute Extrema: For a continuous function, f, on any interval, if x=c is the only critical value in the interval where f (c)=0and f (c) exists, then f(c) is an an on I if f (c)>0 on I if f (c)<0 Note: The test fails if f (c)=0. Example 1: Find two positive numbers whose sum is 60 and whose product is a maximum. Example 2: Find the dimensions of a rectangle of area 225 square centimeters that has the smallest perimeter. What is the perimeter? 8
9 Example 3: Find the dimensions of the rectangle of largest area that has its base on the x-axis and its other two vertices above the x-axis and lying on the parabola y=10 x 2 Example 4: A homeowner has $320 to spend on building a fence around a rectangular garden. Three sides of the fence will be constructed with wire fencing at a cost of $2 per foot. In order to provide a view block for a neighbor, the fourth side is to be constructed with wood fencing at a cost of $6 per foot. Find the dimensions and the area of the largest garden that can be enclosed with $320 worth of fencing. 9
10 Example 5: An open-top box used to carry small toys is to be made from a 10 inch by 12 inch piece of cardboard by cutting identical squares from the corners and then folding up the flaps. Determine the dimensions of the box to maximize the volume. Section 4.6 Highly Suggested Homework Problems: 3, 5, 7, 11, 13 10
11 Section Antiderivatives Example 1: What are the possible functions whose derivative is x? Definition: A function F is called an antiderivative of f on an interval I if F (x)= f(x) for all x in I. Theorem If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is where C is an arbitrary constant. Table of Antidifferentiation Formulas Function General Antiderivative k x n (n 1) 1 x e x a x cosx sinx sec 2 x secxtanx csc 2 x cscxcotx k f(x) f(x)±g(x) 11
12 Example 2: Find the most general antiderivative of each function. a) f(x)=8 b) f(x)=x 5 c) f(x)= 1 3 x4 d) f(t)= 2 t 9 e) f(x)=(5x 4 + x 3 2) f) f(x)= 3 x + sec2 x g) f(x)=e x + x 3 h) f(x)=3 x 1 x 2 x 3 2 i) f(x)= 4x2 + x 3 8x j) f(x)=(x 2)(x+3)+cosx 12
13 Example 3: Find y if y(1)=1 and dy dx = 3 x + 1 x 2 Example 4: Find f(x) if f (x)=5 9x, f(0)=3, and f(1)=10. Example 5: A car is traveling at 70 miles per hour when the brakes are fully applied, producing a constant deceleration of 40 ft/s 2. What is the distance covered before the car comes to a stop? Section 4.8 Highly Suggested Homework Problems: 1, 5, 9, 11, 13, 19, 23, 25, 27, 29, 31, 35, 37, 39, 41, 53 13
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