# The slope of the secant line through the points P ( 3 , f ( 3 ) ) and Q ( x , f ( x ) ) . ### Single Variable Calculus: Concepts...

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

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

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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