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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 the slope of the tangent to the curve y = 1 x the point where x = a.

(b) Find equations of the tangent line at the point (1, 1) and (4, ½).

(c) Graph the curve and both tangent on a common screen.

(a)

To determine

To find: The slope of the tangent line to the curve at the given point.

Explanation

Given:

The equation of the curve is y=1x.

The curve passing through the points (1, 1) and (4,12).

Formula used:

The slope of the tangent curve y=f(x) at the point P(a,f(a)) is,

m=limxaf(x)f(a)xa (1)

Difference of square formula: (a2b2)=(a+b)(ab)

Calculation:

Obtain the slope of the tangent to the curve at the point x=a.

Since f(x)=1x, substitute f(a)=1a in equation (1),

m=limxaf(x)f(a)xa=limxa1x1axa=limxaaxaxxa=limxaax(xa)ax

Multiply both the numerator and the denominator by the conjugate of the numerator

(b)

To determine

To find: The equation of the tangent lines to the curve at the given points.

(c)

To determine

To sketch: The graph of the curve and the tangent lines.

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