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 definition 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 + h)² + 2-(1)² + 2/h
evaluate the limit expression to find f(1).
f(1)=2 (type an integer or a fraction.)
(b) f(1)=3 (type an integer or a fraction.)
the equation of the tangent line at (1, f(1)) is (type your answer in slope - intercept form)

Explanation:

Step1: Recall the point - slope form of a line

The point - slope form of a line is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)\) is a point on the line and \(m\) is the slope of the line. We know that \(x_1 = 1\), \(y_1=f(1)=3\) (from part (b)), and \(m = f^{\prime}(1)=2\) (from part (a)).

Step2: Substitute values into the point - slope form

Substitute \(x_1 = 1\), \(y_1 = 3\), and \(m = 2\) into \(y - y_1=m(x - x_1)\).

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Answer:

\(y = 2x+1\)