Todays Agenda Upcoming Homework Section 4.5: Optimization Problems - - PowerPoint PPT Presentation

Today’s Agenda

• Upcoming Homework
• Section 4.5: Optimization Problems
• Section 4.7: Antiderivatives

Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Mon., 9 November 2015 1 / 7

Upcoming Homework

• WeBWorK HW #20: Section 4.5, due 11/13/2015
• Written HW K: Section 4.3 #16,24,26,34,45. Due 11/13/2015.
• WeBWorK HW #21: Section 4.7, due 11/16/2015

Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Mon., 9 November 2015 2 / 7

Section 4.5

Optimization problems are those that allow us to maximize or minimize a certain quantity. The following steps provide an outline of how to solve an

• ptimization problem:

1 Understand the problem. Critically read the problem statement, and

determine the knowns, unknowns, and given conditions.

2 Draw a diagram. 3 Introduce notation. Also label the diagram with the notation you

have chosen.

4 Express the quantity that you are trying to maximize or minimize in

terms of the other variables in the problem.

5 Find the absolute maximum or minimum value of the quantity in

question.

Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Mon., 9 November 2015 3 / 7

Section 4.5

Practice Problems

1 Find the area of the largest rectangle that can be inscribed in a

semicircle of radius r.

2 Find the point on the parabola y2 = 2x that is closest to the point

(1, 4).

3 A man launches his boat from point A on a bank of a straight river, 3

km wide, and wants to reach point B, 8 km downstream on the

• pposite bank, as quickly as possible. He could row his boat directly

across the river to point C and then run to B, or he could row directly to B, or he could row to some point D between C and B and then run to B. If he can row 6 km/hr and run 8 km/hr, where should he land to reach B as soon as possible? (We assume that the speed of the water is negligible compared with the speed at which the man rows.)

Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Mon., 9 November 2015 4 / 7

Section 4.7

For the majority of the semester, we have been concerned with taking

• derivatives. Now we will discuss how to ”undo” the process of

differentiation.

Definition 4.7.1

A function F is called an antiderivative of the function f on an interval I if F ′(x) = f (x) for all x ∈ I.

Theorem 4.7.2

If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is F(x) + C, where C is an arbitrary constant. (The set {F(x) + C | C ∈ R} is often called the family of antiderivatives for f .)

Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Mon., 9 November 2015 5 / 7

Section 4.7

Let’s find general antiderivatives for each of the following common functions:

1 f (x) = xn 2 f (x) = 1

x

3 f (x) = ex 4 f (x) = cos x 5 f (x) = sin x 6 f (x) = sec2 x 7 f (x) = sec x tan x 8 f (x) =

1 √ 1 − x2

9 f (x) =

1 1 + x2

Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Mon., 9 November 2015 6 / 7

Section 4.7

Complete the following problems with a partner. Find a general antiderivative of each of the following functions:

1 f (x) = 8x9 − 3x6 + 12x3 2 f (x) = 3

√ x2 + x√x

3 f (t) = 3t4 − t3 + 6t2

t4

4 f (θ) = sec θ tan θ − 2eθ 5 f (x) = 2√x + 6 cos x 6 f (x) = 2 + x2

1 + x2 (Hint: write the numerator as 1 + (1 + x2).)

Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Mon., 9 November 2015 7 / 7