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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

(a) Find an equation of the tangent to the curve y = ex that is parallel to the line x – 4y = 1.

(b) Find an equation of the tangent to the curve y = ex that passes through the origin.

(a)

To determine

To find: The equation of the tangent line to the curve that is parallel to the line x4y=1.

Explanation

Given:

The curve is y=ex and the line is x4y=1.

Formula Used:

The equation of the tangent line at (x1,y1) is, yy1=m(xx1) (1)

Where, m is the slope of the tangent line at (x1,y1) and m=dydx|x=x1.

Calculation:

Step 1:

Obtain the slope of the tangent line to the curve.

The derivative of curve y is as follows.

dydx=ddx(ex)=ex

Thus, the derivative of curve y is ex.

Since the derivative of the curve is the slope of the tangent to the curve, the slope of the tangent to the curve y=ex is ex.

Step 2:

Obtain the tangent point.

Express the given line as, y=14x14 and its slope is 14.

Since the required equation of the tangent line is parallel to the given line, the slope of the tangent line is equal to the slope of the given line.

That is, ex=14.

Solve the equation ex=14 and obtain the value of x.

ex=14

Take natural logarithm on both sides,

lnex=ln(14)x=ln(14)

Substitute ln(14) for x in y=ex and obtain the value of y

(b)

To determine

To find: The equation of the tangent line to the curve which is passes through the origin.

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