Calculus I (MAT 145) Dr. Day Wednesday March 27, 2019 Chapter 4: Using All Your Derivative Knowledge! Absolute and Relative Extremes What is a “critical number?” Increasing and Decreasing Behavior of Functions Connecting f and f’ Concavity of Functions: A function’s curvature Connecting f and f” Graphing a Function: Putting it All together! Max-Mins Problems: Determine Solutions for Contextual Situations Other Applications Finally . . . What if We Reverse the Derivative Process? Test #3 Friday! STV 346 Wednesday, March 27, 2019 MAT 145

What does f’ tell us about f? If f’(c)= 0, there is a horizontal tangent to the curve at x=c. This may mean there is a local max or min at x=c. If f’(c) is undefined, there could be a discontinuity, a vertical tangent, or a cusp (sharp point) at x=c. If f(x) is continuous at x=c, there may be a local max or min at x=c. Wednesday, March 27, 2019 MAT 145

First derivative test Wednesday, March 27, 2019 MAT 145

Concavity of a Function Concavity Animations More Concavity Animations Wednesday, March 27, 2019 MAT 145

Concavity of a Function Concavity Animations More Concavity Animations Wednesday, March 27, 2019 MAT 145

What does f’’ tell us about f? If f’’(c)> 0, then the original curve f(x) is concave up at x=c. If f’’(c)< 0, then the original curve f(x) is concave down at x=c. If f’’(c)= 0, then f(x) is neither concave up nor concave down at x=c. And there could be an inflection point on f(x) at x=c. If f’’(c) is undefined, there could be a discontinuity, a vertical tangent, or a cusp (sharp point) in f’(x) at x=c. There may be a change of concavity in f(x) at x=c. Wednesday, March 27, 2019 MAT 145

Inflection Point Wednesday, March 27, 2019 MAT 145

Info about f from f ’ Here’s a graph of g’(x). Determine all intervals over which g is increasing and over which g is decreasing. Identify and justify where all local extremes occur. Wednesday, March 27, 2019 MAT 145

Info about f from f ’’ Here’s a graph of h”(x). Determine all intervals over which h is concave up and over which h is concave down. Identify and justify where all points of inflection occur. Wednesday, March 27, 2019 MAT 145

Optimization What does it mean to optimize? What are examples in which you might want to determine an optimum solution? What should you consider when looking for an optimum solution? Wednesday, March 27, 2019 MAT 145

Open top box A square piece of cardboard 1 foot on each side is to be cut and folded into a box with no top. To accomplish this, congruent squares will be cut from each corner of the cardboard. Determine the dimensions of the box that will enclose the greatest volume. Wednesday, March 27, 2019 MAT 145

Optimization: An example A farmer has 2400 ft of fencing and wants to fence off a rectangular field that borders a straight river. She needs no fence along the river. What are the dimensions of the field that has the largest area? What are we trying to optimize here and in what way (greatest or least)? Which variables are involved? What constraints should be taken into account? What initial guesses do you have about a solution? Wednesday, March 27, 2019 MAT 145

Optimization: Guess & check A farmer has 2400 ft of fencing and wants to fence off a rectangular field that borders a straight river. She needs no fence along the river. What are the dimensions of the field that has the largest area? What are options? Guess and check to fit constraints. Are these all the options? Wednesday, March 27, 2019 MAT 145

Optimization: Calculus strategy A farmer has 2400 ft of fencing and wants to fence off a rectangular field that borders a straight river. She needs no fence along the river. What are the dimensions of the field that has the largest area? Let x and y be the length and width of the rectangular field (in feet). Then we express A, area of rectangular field (in sq. feet) in terms of x and y: A = xy Optimization Equation: A=xy Constraint: 2x + y = 2400 Combine: A = x(2400 – 2x) = 2400x – 2x2 Domain: x is within [0, 1200] Now, find absolute maximum of A. (Just like we did when graphing.) Wednesday, March 27, 2019 MAT 145

Optimize Equation to maximize: A (x) = 2400x – 2x2 , A represents area of field in sq. ft; x represents width of field, as shown in the diagram. Domain: 0  x  1200; Thus, endpoints to check are x=0, x=1200. Find derivative and critical numbers (locations at which derivative is zero or undefined): A (x) = 2400 – 4x 2400 – 4x = 0 x = 600 The absolute maximum value of A must occur either at this critical number or at an endpoint of the interval. Since A(0) = 0, A(600) = 720,000, and A(1200) = 0, the Closed Interval Method gives the maximum value as A(600) = 720,000. Dimensions of field with greatest area: 600 ft by 1200 ft. Note: A’ function is a polynomial; so A’ is defined everywhere on its domain. Wednesday, March 27, 2019 MAT 145

Steps for Optimizing Understand the problem: What is given? What is requested? Are there constraints? Diagram and variables: If appropriate for the context draw a diagram and label with variables. Define variables (by saying what the letters you are using represent, including units) and determine a reasonable span of values for those variables in the context of the problem. This will help you identify the domain of the independent variable. Determine quantity to be optimized and write an equation: Read the problem to determine what you have been asked to optimize. Express the quantity to be optimized (one of your variables), as a function of the other variable(s) in the problem. Identify constraint equation, if appropriate: If there are several variables, one may be constrained to be a function of the others. Write an equation that relates those variables. Combine with other equation previously defined, if appropriate. Use the closed interval method to identify the absolute maximum or minimum Answer the question! Wednesday, March 27, 2019 MAT 145

Bendable Wire A bendable wire measures 1000 cm in length. Write a function, call it A(x), to represent the sum of the areas of a circle and a square that result when that wire is cut at one point and the resulting two pieces of wire are used to create those shapes. Determine how to cut the wire so that the sum of the areas of the circle and the wire are a maximum. Wednesday, March 27, 2019 MAT 145

Bendable Wire Wednesday, March 27, 2019 MAT 145

Bendable Wire Note: The A’ function is a polynomial; so A’ is defined everywhere on its domain. Thus, the only critical numbers occur when A’ = 0. A(4000/(π+4)) ≈35,006, A(0)≈79,577, A(1000)=62,500. Thus, max area occurs when x=0 and entire wire is used to make a circle. Wednesday, March 27, 2019 MAT 145

Solve the follow optimization problem Solve the follow optimization problem. Show complete evidence and calculus justification. Include a drawing or a graph to represent the situation. Include evidence that shows you have considered any domain restrictions.   ____ 1 pt: labeled sketch, drawing, graph; ____ 1 pt: variables identified/described; ____ 1 pt: domain of independent variable; ____ 1 pt: statement of optimizing function; ____ 1 pt: constraint; ____ 1 pt: calculus evidence; ____ 1 pt: justify optimum; ____ 1 pt: consider all possibilities for critical points; ____ 1 pt: correct solution, units labeled Wednesday, March 27, 2019 MAT 145