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Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

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BuyFindarrow_forward

Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

Linear Approximation Consider the function f ( x ) = x 3 / 2 with the solution point (4.8).

(a) Use a graphing utility to graph f. Use the zoom feature to obtain successive magnifications of the graph in the neighborhood of the point (4. 8). After zooming in a few times, the graph should appear nearly linear. Use the trace feature to determine the coordinates of a point near (4, 8). Find an equation of the secant line S(x) through the two points.

(b) Find the equation of the line T ( x ) = f ' ( 4 ) ( x 4 ) + f ( 4 ) tangent to the graph of f passing through the given point. Why are the linear functions S and T nearly the same?

(c) Use a graphing utility to graph f and T on the same set of coordinate axes. Note that T is a good approximation of f when x is close to 4. What happens to the accuracy of the approximation as you move farther away from the point of tangency?

Δ x -3 -2 -1 -0 5 -0.1 0
f ( 4 + Δ x )
T ( 4 + Δ x )
Δ x 0.1 0.5 1 2 3
f ( 4 + Δ x )
T ( 4 + Δ x )

(a)

To determine

To Find: The equation of the secant line.

Explanation

Given:

f(x)=x3/2

Formula used:

m=y2y1x2x1

Calculation:

A point in the neighborhood of (4, 8) is (3, 5.2), the two are plotted on the graph below.

Equation of the secant line S(x) through the two points is the equation of a line passing through the two points

(b)

To determine

To Find: The equation of the tangent line.

(c)

To determine

To graph: Graph f and T.

(d)

To determine

To Compute: The conclusion

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P-2ESect-3.8 P-3ESect-3.8 P-4ESect-3.8 P-5ESect-3.8 P-6ESect-3.8 P-7ESect-3.8 P-8ESect-3.8 P-9ESect-3.8 P-10ESect-3.8 P-11ESect-3.8 P-12ESect-3.8 P-13ESect-3.8 P-14ESect-3.8 P-15ESect-3.8 P-16ESect-3.8 P-17ESect-3.8 P-18ESect-3.8 P-19ESect-3.8 P-20ESect-3.8 P-21ESect-3.8 P-22ESect-3.8 P-23ESect-3.8 P-24ESect-3.8 P-25ESect-3.8 P-26ESect-3.8 P-27ESect-3.8 P-28ESect-3.8 P-29ESect-3.8 P-30ESect-3.8 P-31ESect-3.8 P-32ESect-3.8 P-33ESect-3.8 P-34ESect-3.8 P-35ESect-3.8 P-36ESect-3.8 P-37ESect-3.8 P-38ESect-3.8 P-39ESect-3.8 P-40ESect-3.8 P-41ESect-3.8 P-42ESect-3.8 P-43ESect-3.8 P-44ESect-3.8 P-45ESect-3.8 P-46ECh-3 P-1RECh-3 P-2RECh-3 P-3RECh-3 P-4RECh-3 P-5RECh-3 P-6RECh-3 P-7RECh-3 P-8RECh-3 P-9RECh-3 P-10RECh-3 P-11RECh-3 P-12RECh-3 P-13RECh-3 P-14RECh-3 P-15RECh-3 P-16RECh-3 P-17RECh-3 P-18RECh-3 P-19RECh-3 P-20RECh-3 P-21RECh-3 P-22RECh-3 P-23RECh-3 P-24RECh-3 P-25RECh-3 P-26RECh-3 P-27RECh-3 P-28RECh-3 P-29RECh-3 P-30RECh-3 P-31RECh-3 P-32RECh-3 P-33RECh-3 P-34RECh-3 P-35RECh-3 P-36RECh-3 P-37RECh-3 P-38RECh-3 P-39RECh-3 P-40RECh-3 P-41RECh-3 P-42RECh-3 P-43RECh-3 P-44RECh-3 P-45RECh-3 P-46RECh-3 P-47RECh-3 P-48RECh-3 P-49RECh-3 P-50RECh-3 P-51RECh-3 P-52RECh-3 P-53RECh-3 P-54RECh-3 P-55RECh-3 P-56RECh-3 P-57RECh-3 P-58RECh-3 P-59RECh-3 P-60RECh-3 P-61RECh-3 P-62RECh-3 P-63RECh-3 P-64RECh-3 P-65RECh-3 P-66RECh-3 P-67RECh-3 P-68RECh-3 P-69RECh-3 P-70RECh-3 P-71RECh-3 P-72RECh-3 P-73RECh-3 P-74RECh-3 P-75RECh-3 P-76RECh-3 P-77RECh-3 P-78RECh-3 P-79RECh-3 P-80RECh-3 P-81RECh-3 P-82RECh-3 P-83RECh-3 P-84RECh-3 P-85RECh-3 P-86RECh-3 P-87RECh-3 P-88RECh-3 P-89RECh-3 P-90RECh-3 P-91RECh-3 P-92RECh-3 P-93RECh-3 P-94RECh-3 P-95RECh-3 P-96RECh-3 P-97RECh-3 P-98RECh-3 P-99RECh-3 P-100RECh-3 P-101RECh-3 P-102RECh-3 P-103RECh-3 P-104RECh-3 P-105RECh-3 P-106RECh-3 P-107RECh-3 P-108RECh-3 P-109RECh-3 P-110RECh-3 P-111RECh-3 P-112RECh-3 P-113RECh-3 P-114RECh-3 P-115RECh-3 P-116RECh-3 P-117RECh-3 P-118RECh-3 P-119RECh-3 P-120RECh-3 P-121RECh-3 P-122RECh-3 P-123RECh-3 P-124RECh-3 P-125RECh-3 P-126RECh-3 P-127RECh-3 P-128RECh-3 P-129RECh-3 P-130RECh-3 P-1PSCh-3 P-2PSCh-3 P-3PSCh-3 P-4PSCh-3 P-5PSCh-3 P-6PSCh-3 P-7PSCh-3 P-8PSCh-3 P-9PSCh-3 P-10PSCh-3 P-11PSCh-3 P-12PSCh-3 P-13PSCh-3 P-14PSCh-3 P-15PSCh-3 P-16PS

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