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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

The point P(0.5, 0) lies on the curve y = cos πx.

(a) If Q is the point (x, cos πx), use your calculator to find the slope of the secant line PQ (.correct to six decimal places) for the following values of x:

(i) 0

(ii) 0.4

(iii) 0.49

(iv) 0.499

(v) 1

(vi) 0.6

(vii) 0.51

(viii) 0.501

(b) Using the result of part (a), guess the value of the slope of the tangent line to the curve at P(0.5, 0).

(c) Using the slope from part (b), find an equation of the tangent line to the curve at P(0.5, 0).

(d) Sketch the curve, two of the secant lines, and the tangent line.

(a)

To determine

To find: The slope of the secant line PQ for the following values of x.

Explanation

Given:

The equation of the curve y=cosπx.

The point P(0.5, 0) lies on the curve y.

The point Q is (x,cosπx).

Calculation:

The slope of the secant line between the points, P(0.5, 0) and Q(x,cosπx) is

mPQ=cosπx0x0.5 (1)

Section (i):

Obtain the slope of the secant line PQ when x=0.

Substitute 0 for x in cosπx.

cosπx=cosπ(0)=1

Substitute Q(x,cosπx)=(0,1) in equation (1).

mPQ=cosπx0x0.5=1000.5=10.5=2

Thus, the slope of the secant line PQ when x=0 is 2.

Section-(ii):

Obtain the slope of the secant line PQ when x=0.4.

Substitute 0.4 for x in cosπx.

cosπx=cosπ(0.4)=0.309016990.309017

Substitute Q(x,cosπx)=(0.4,0.309017) in equation (1).

mPQ=cosπx0x0.5=0.30901700.40.5=0.3090170.13.090170

Thus, the slope of the secant line PQ when x=0.4 is 3.090170.

Section-(iii):

Obtain the slope of the secant line PQ when x=0.49.

Substitute 0.49 for x in cosπx.

cosπx=cosπ(0.49)=0.03141070.031411

Substitute Q(x,cosπx)=(0.49,0.031411) in equation (1).

mPQ=cosπx0x0.5=0.03141100.490.5=0.0314110.013.141076

Thus, the slope of the secant line PQ when x=0.49 is 3.141076.

Section-(iv):

Obtain the slope of the secant line PQ when x=0.499.

Substitute 0.499 for x in cosπx.

cosπx=cosπ(0.499)=0.0031415870.00314159

Substitute Q(x,cosπx)=(0.499,0.00314159) in equation (1).

mPQ=cosπx0x0.5=0.0031415900.4990.5=0.003141590.0013

(b)

To determine

To guess: The slope of the tangent line to the curve at P (0.5, 0).

(c)

To determine

To find: The equation of the tangent line to the curve at P(0.5, 0).

(d)

To determine

To sketch: The curve, two of the secant line, and the tangent line.

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