Question

use the function below to answer parts (a)-(c).
f(x)=x² + 2
(a) use the formal definition to find the derivative of y = f(x) at x = 1.
(b) find f(1) and find the equation of the tangent line at the point (1, f(1)).
(c) graph y = f(x) and the tangent line at the point (1, f(1)) in the same coordinate system.
(a) the derivative of a function f at x, denoted by f(x), is f(x) = limₕ→0 f(x + h)-f(x)/h provided that the limit exists. use the
artially correct
of the derivative of f at x to find the derivative of the given function f(x) when x = 1.
f(1) = limₕ→0 □ - (1)² + 2/h

Explanation:

Step1: Find \(f(x + h)\)

Given \(f(x)=x^{2}+2\), then \(f(x + h)=(x + h)^{2}+2=x^{2}+2xh+h^{2}+2\)
When \(x = 1\), \(f(1 + h)=(1 + h)^{2}+2=1 + 2h+h^{2}+2=h^{2}+2h + 3\)

Step2: Substitute into the derivative formula

$$
LATEXBLOCK0
$$

Step3: Evaluate the limit

As \(h
ightarrow0\), \(\lim_{h
ightarrow0}(h + 2)=2\)

Answer:

The value of \(f^{\prime}(1)\) is \(2\)