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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 3: Q. 20 (page 166)

Calculus: Simpson鈥檚 Rule The figure shows the graph of $y = a x^{2} + b x + c$ . Suppose that the points localid="1647353664782" $(- h, y_{0}), (0, y) a n d (h, y_{2})$ are on the graph. It can be shown that the area enclosed by the parabola, the localid="1647353732361" $x$ -axis, and the lineslocalid="1647353669382" $x = - h, x = h$ is Area localid="1647353674576" $= \frac{h}{3} (2 a h^{2} + 6 c) .$ Show that this area may also be given by Area localid="1647353678398" $= \frac{h}{3} (y_{0} + 4 y_{1} + y_{2}) .$

Short Answer

Expert verified

$y = a x^{2} + b x + c$ It is proven that arealocalid="1647353590732" $= \frac{h}{3} (y_{0} + 4 y_{1} + y_{2}) .$

Step by step solution

01

Step 1. Given information

The given figure is,

The figure shows the graph $y = a x^{2} + b x + c .$ Suppose that the point $(- h, y_{0}), (0, y) a n d (h, y_{2})$ is on the graph and the area is an area $= \frac{h}{3} (2 a h^{2} + 6 c) .$ We need to show that the area may be given by Area $= \frac{h}{3} (y_{0} + 4 y_{1} + y_{2}) .$

02

Step 2. Simplify

The figure is $y = a x^{2} + b x + c .$

For the point $(- h, y_{0}),$ the function is,

$y_{0} = a 路 {(- h)}^{2} + b (- h) + c .$

role="math" localid="1647352990982" $y_{0} = h^{2} - b h + c 鈰� (1) .$

For the point $(0, y_{1}),$ the function is,

$y_{1} = c 鈰� (2) .$

For the point $(h, y_{2})$ the function is,

$y_{2} = a h^{2} + b h + c 鈰� (3) .$

Add equations (1),(3) and include equation (2).

$y_{0} + y_{2} = 2 路 a 路 h^{2} + 2 路 y_{1} .$

$2 路 a 路 h^{2} = y_{0} + y_{2} - 2 y_{1} 鈰� (4) .$

Now include equation (4) in the equation of the area.

Area role="math" localid="1647353241247" $= \frac{h}{3} (2 a h^{2} + 6 c) .$

$= \frac{h}{3} (y_{0} + 4 y_{1} - 2 y_{1} + 6 y_{1}) .$

$= \frac{h}{3} (y_{0} + 4 y_{1} + y_{2}) .$

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Most popular questions from this chapter

Constructing Rain Gutters A rain gutter is to be made of aluminum sheets that are 12 inches wide by turning up the edges 90. See the illustration.

(a) What depth will provide maximum cross-sectional area and hence allow the most water to flow?

(b) What depths will allow at least 16 square inches of water to flow?

The price p (in dollars) and the quantity x sold of a certain product obey the demand equation $p = - \frac{1}{10} x + 1000$ .

(a) Find a model that expresses the revenue R as a function of x.

(b) What is the revenue if 400 units are sold?

(c) What quantity x maximizes revenue? What is the maximum revenue?

(d) What price should the company charge to maximize revenue?

The marginal cost of a product can be thought of as the cost of producing one additional unit of output. For example, if the marginal cost of producing the 50th product is \(6.20, it cost \)6.20 to increase production from 49 to 50 units of output. Suppose the marginal cost C

(in dollars) to produce x thousand mp3 players is given by the function

$f (x) = x^{2} - 140 x + 7400$

(a) How many players should be produced to minimize the marginal cost?

(b) What is the minimum marginal cost?

Enclosing the Most Area with a Fence: A farmer with 4000 meters of fencing wants to enclose a rectangular plot that borders on a river. If the farmer does not fence the side along the river, what is the largest area that can be enclosed? ( See the figure)

Graph the function $f$ by starting with the graph of $y = x^{2}$ and using transformations (shifting, compressing, stretching, and/or reflection). Verify your results using a graphing utility.

Hint: If necessary, write $f$ in the form $f (x) = a (x - h)^{2} + k$ .

$f (x) = 3 x^{2} + 6 x$

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