BuyFind

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 2.6, Problem 1E

(a)

To determine

To write: The slope of the secant line through the points P(3,f(3)) and Q(x,f(x)).

Expert Solution

Answer to Problem 1E

The slope of the secant line through the points P(3,f(3)) and Q(x,f(x)) is mPQ=f(x)f(3)x3_.

Explanation of Solution

Given:

The equation of the curve y=f(x).

The slope of the secant line through the points P(3,f(3)) and Q(x,f(x)).

Formula used:

The slope of the secant line through the points P(a,f(a)) and Q(x,f(x)) is,

mPQ=f(x)f(a)xa . (1)

Calculation:

Obtain the slope of the secant line through the points P(3,f(3)) and Q(x,f(x)).

Substitute 3 for a in equation (1),

mPQ=f(x)f(3)x3

Thus, the slope of the secant line mPQ=f(x)f(3)x3_.

(b)

To determine

To write: The slope of the tangent line at P.

Expert Solution

Answer to Problem 1E

The slope of the tangent line at P is limx3f(x)f(3)x3_.

Explanation of Solution

Given:

The equation of the curve y=f(x).

The slope of the secant line through the points P(3,f(3)) and Q(x,f(x)).

Formula used:

The slope of the tangent line at the point P(a,f(a)) is,

m=limxaf(x)f(a)xa (2)

Calculation:

The slope of the tangent line at the point P is the limit of the slope of the secant line PQ as Q approaches P.

From part (a), the slope of the secant line at PQ is, mPQ=f(x)f(3)x3.

Substitute mPQ for f(x)f(a)xa in equation (2),

m=limx3mPQ=limx3f(x)f(3)x3

Thus, the slope of the tangent line at the point P is m=limx3f(x)f(3)x3_.

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