Maximum and Minimum Values What is the shape of a can that minimizes manufacturing costs? What is the maximum acceleration of a space shuttle? (This is an important question to the astronauts who have to withstand the effects of acceleration.) What is the radius of a contracted windpipe that expels air most rapidly during a cough? At what angle should blood vessels branch so as to minimize the energy expended by the heart in pumping blood?

What Does the First Derivative Function Say about the Original Function

What Does the Second Derivative Function Say about the Original Function

Indeterminate Forms and L’Hospital’s Rule

Summary of Curve Sketching The following checklist is intended as a guide to sketching a curve by hand. 1.Find the Domain of the Function. 2.Find y-Intercepts and x-Intercepts. 3. Find all the Asymptotes of the Function..

4. Find out the Intervals of Increase or Decrease. 5. Find out the Local Maximum and Minimum Points. 6. Find out the Concavity and Points of Inflection. 7.Analyze the Function in a Table. 8.Sketch the Curve; Using the Information in Items Sketch the asymptotes as dashed lines. Plot the intercepts, maximum and minimum points, and inflection points. Then make the curve pass through these points, rising and falling according to 4, with concavity according to 6, and approaching the asymptotes.

Optimization Problems A businessperson wants to minimize costs and maximize profits. A traveler wants to minimize transportation time. Fermat’s Principle in optics states that light follows the path that takes the least time. A farmer wants to have a filed which has the largest area but he has fixed limited of fencing to fence the filed. We are going to solve such problems as maximizing areas, volumes, and profits and minimizing distances, times, and costs.

Steps in Solving Optimization Problems 1. Understand the Problem The first step is to read the problem carefully until it is clearly understood. Ask yourself: What is the unknown? What are the given quantities? What are the given conditions? 2. Draw a Diagram In most problems it is useful to draw a diagram and identify the given and required quantities on the diagram. 3. Introduce Notation Assign a symbol to the quantity that is to be maximized or minimized (let’s call it Q ). Also select symbols a,b,c,… for other unknown quantities and label the diagram with these symbols. It may help to use initials as suggestive symbols for example, A for area, h for height, t for time.

4.Express Q in terms of some of the other symbols. If Q has been expressed as a function of more than one variable, use the given information to find relationships (in the form of equations) among these variables. Then use these equations to eliminate all but one of the variables in the expression for Q. Thus, Q will be expressed as a function of one variable x, say, 5. Use the derivative function to find the absolute maximum or minimum value of f. If the domain of f is a closed interval, then the Closed Interval Method in Section 4.1 can be used.

Newton’s Method

Antiderivatives