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Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 2.1, Problem 4E

(a)

To determine

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

Expert Solution

Answer to Problem 4E

The slope of the secant line PQ for the following values of x is given below:

(i) The slope of the secant line PQ when x=0 is 2.

(ii) The slope of the secant line PQ when x=0.4 is 3.090170.

(iii) The slope of the secant line PQ when x=0.49 is 3.141076.

(iv) The slope of the secant line PQ when x=0.499 is 3.141587.

(v) The slope of the secant line PQ when x=1 is 2.

(vi) The slope of the secant line PQ when x=0.6 is 3.090170.

(vii) The slope of the secant line PQ when x=0.51 is 3.141076.

(viii) The slope of the secant line PQ when x=0.501 is 3.141587.

Explanation of Solution

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=0.003141590.0013.141587

Thus, the slope of the secant line PQ when x=0.499 is 3.141587.

Section-(v):

Obtain the slope of the secant line PQ when x=1.

Substitute 1 for x in cosπx.

cosπx=cosπ(1)=cosπ=1

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

mPQ=cosπx0x0.5=1010.5=10.52

Thus, the slope of the secant line PQ when x=1 is 2.

Section-(vi):

Obtain the slope of the secant line PQ when x=0.6.

Substitute 0.6 for x in cosπx.

cosπx=cosπ(0.6)=0.309016990.309017

Substitute Q(x,cosπx)=(0.6,0.309017) in equation (1).

mPQ=cosπx0x0.5=0.30901700.60.5=0.3090170.13.090170

Thus, the slope of the secant line PQ when x=0.6 is 3.090170.

Section-(vii):

Obtain the slope of the secant line PQ when x=0.51.

Substitute 0.49 for x in cosπx.

cosπx=cosπ(0.51)=0.03141070.031411

Substitute Q(x,cosπx)=(0.51,0.031411) in equation (1).

mPQ=cosπx0x0.5=0.03141100.510.5=0.0314110.013.141076

Thus, the slope of the secant line PQ when x=0.51 is 3.141076.

Section-(viii):

Obtain the slope of the secant line PQ when x=0.501.

Substitute 0.501 for x in cosπx.

cosπx=cosπ(0.501)=0.0031415870.00314159

Substitute Q(x,cosπx)=(0.501,0.00314159) in equation (1).

mPQ=cosπx0x0.5=0.0031415900.5010.5=0.003141590.0013.141587

Thus, the slope of the secant line PQ when x=0.501 is 3.141587.

(b)

To determine

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

Expert Solution

Answer to Problem 4E

The estimated slope of the tangent line to the curve at P (0.5, 0) is π.

Explanation of Solution

Formula used:

The slope of the tangent line is the limit of the slope of the secant line.

That is, limQP(mPQ)=m (2)

Calculation:

From part (a), the slope of the secant line for many values of x is closer to 1. Thus, the slope mPQ approaches to 3.142 or π .

Substitute mPQ=cosπx0x0.5 in equation (2).

m=limQP(mPQ)=limx0.5(cosπx0x0.5)=(cosπ(0.5)0.50.5)=00

Since 00 is in indeterminate form, apply L’Hospital’s rule and obtain the limit.

limx0.5(cosπx0x0.5)=limx0.5(πsin(πx))=(πsin(π(0.5)))=π(1)=π

Thus, the estimated slope of the tangent line to the curve at P (0.5, 0) is π.

(c)

To determine

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

Expert Solution

Answer to Problem 4E

The equation of the tangent line to the curve at P(0.5, 0) is y=πx+12π_

Explanation of Solution

Formula used:

The equation of the tangent line to the curve y=f(x) at the point (x1,y1):

yy1=m(xx1) (3)

Calculation:

Substitute m=π and (x1,y1)=(0.5,0) in equation (3).

y0=π(x0.5)y=πx+0.5π=πx+12π

Thus, the equation of the tangent line to the curve at P(0.5,0) is y=πx+12π_.

(d)

To determine

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

Expert Solution

Explanation of Solution

Formula used:

Slope-intercept formula: yy1=mPQ(xx1)

Calculation:

Obtain the secant line at x=0.

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

Use the slope-point intercept formula to find a equation of the secant line.

Substitute (x1,y1)=(0,1) and mPQ=2 in the Slope-point intercept formula.

y1=2(x0)y1=2xy=2x+1

Thus, the equation of the secant line at x=0 is y=2x+1.

Obtain the secant line at x=1.

The slope of the secant line PQ when x=1 is 2.

Use the slope-point intercept formula to find a equation of the secant line.

Substitute (x1,y1)=(1,1) and mPQ=2 in the Slope-point intercept formula.

y(1)=2(x1)y+1=2x+2y=2x+21=2x+1

Thus, the equation of the secant line at x=1 is y=2x+1.

Note that the equation of the secant line at x=0 and x=1 are the same.

Draw the curve y=cosπx , two of the secant lines y=2x+1, and the tangent line y=πx+π2 as shown below in Figure 1.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 2.1, Problem 4E

Thus, the required sketch is obtained.

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