Q. 8 - Calculus

Question

On a graph of f(x) = $x^2$,

(a) draw the tangent line to the graph of f at the point (2, f(2)); 

(b) draw the secant line from (2, f(2)) to (2.75, f(2.75)); 

(c) draw the secant line from (1.75, f(1.75)) to (2, f(2)). 

(d) Which secant line is a better approximation to the tangent line, and why? 

Step-by-Step Solution

Answer

Part (a).

Part (b).

Part (c).

Part (d). Secant line from (1.75, f(1.75)) to (2, f(2)  is a better approximation to the tangent line.

Part (a) Step 1. Given information.

We have to draw the tangent line to the graph of f at the point (2, f(2));

Draw the secant line from (2, f(2)) to (2.75, f(2.75));  draw the secant line from (1.75, f(1.75)) to (2, f(2)). 

Part (a) Step 2. Draw the graph of f ( x ) = x 2 and a tangent line to this graph at the point (2, f(2)).

Prepare the table of values:

x	f(x)
- 2	4
- 1	1
0	0
1	1
2	4

The graph of the function is:

Find f(2).

$$\begin{gathered} f\left(2\right) \\ =2^2 \\ =4 \end{gathered}$$

Part (b) Step 1. Draw the secant line from (2, f(2)) to (2.75, f(2.75))

Find the value of f(2.75).

$$\begin{gathered} f\left(2.75\right)={2.75}^2 \\ =7.5625 \end{gathered}$$

Part (c) Step 1. Draw the secant line from (1.75, f(1.75)) to (2, f(2)).

Find the value of f(1.75).

Part (d) Step 1. Explain which secant line is a better approximation to the tangent line, and why.

In the graph, we can see that the secant line for the points (1.75, f(1.75)) and (2, f(2)) gives a better approximation than the secant line for the points (2, f(2)) and (2.75, f(2.75)). This is because 2 is close to 1.75 than 2.75.