Secant lines

Question

Consider the function f(x) = sqrt (x) and the point P(4, 2) on the graph of f.

(a) Graph f and the secant lines passing through P(4, 2) and Q(x, f(x)) for x- values of 1, 3 and 5.

(b) Find the slope of each secant lines.

(c) Use the results of part (b) to estimate the slope of the tangent line to the graph of f at P(4, 2).

Describe how to improve your approximation of the slope.

Answer 1

(b)

Step 1:

The secant lines are , , and .

The slopes of the secant lines are .

 

Solution :

The slopes of the secant lines are .

Answer 2

(c)

Step 1 :

The function is

The slope of the tangent line  at is .

At

Step 2 :

Evaluating the limits.

.

Apply L-hospital rule.

.

Evaluating the limits.

The slope of the tangent line is .

 

Solution :

The slope of the tangent line is .

Answer 3

(a)

Step 1:

The function is and points and .

At ,

At , and

At .

The Q points are :

Step 2 : 

First find the secant line passing through the points and .

Let and .

Slope intercept form of the line equation where  m is slope and b is y-intercept.

The line equation PQ is

Find the y- intercept by substituting any point on line PQ , say .

The line equation PQ is .

Step 3: 

Find the secant line passing through the points and .

Let and .

Slope intercept form of the line equation where  m is slope and b is y-intercept.

The line equation PQ is

Find the y- intercept by substituting any point on line PQ , say .

The line equation PQ is .

Answer 4

Contd............

Step 4: 

Find the secant line passing through the points and .

Let and .

Slope intercept form of the line equation where  m is slope and b is y-intercept.

The line equation PQ is

Find the y- intercept by substituting any point on line PQ , say .

The line equation PQ is .

The secant lines are , , and .

Step 5:

The graph of the function and the secant lines is :

Solution :

 (a) The graph of the function and the secant lines is :