Chapter 3.3, Problem 2QQ

To determine

The correct option which gives the value of $f^{\prime }\left(1\right)$ .

Answer to Problem 2QQ

The correct option is (A).

Explanation of Solution

Given information:

The points: $\left(1,2\right)$ , $\left(-1,1\right)$

The options:

(A) $-2$

(B) 2

(C) $-\frac{1}{2}$

(D) $\frac{1}{2}$

(E) 3

Calculation:

Given that the normal line passes through $\left(-1,1\right)$ and $\left(1,2\right)$ .

Here, the slope of normal line is:

  $\frac{2-1}{1-\left(-1\right)}=\frac{1}{2}$

And, the slope of the tangent line:

  $-\frac{1}{\frac{1}{2}}=-2$

It is seen that the slope of the tangent is negative reciprocal of the slope of the normal line.

This means that $f^{\prime }\left(1\right)=-2$ .

Hence, the correct option is (A).