use the function below to answer parts (a)-(c).\nf(x)=x² + 2\n(a) use the formal definition to find the…

use the function below to answer parts (a)-(c).\nf(x)=x² + 2\n(a) use the formal definition to find the derivative of y = f(x) at x = 1.\n(b) find f(1) and find the equation of the tangent line at the point (1, f(1)).\n(c) graph y = f(x) and the tangent line at the point (1, f(1)) in the same coordinate system.\n(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.\nf(1)=limₕ→0 (1 + h)² + 2-(1)² + 2/h\nevaluate the limit expression to find f(1).\nf(1)=2 (type an integer or a fraction.)\n(b) f(1)=3 (type an integer or a fraction.)\nthe equation of the tangent line at (1, f(1)) is (type your answer in slope - intercept form)

use the function below to answer parts (a)-(c).\nf(x)=x² + 2\n(a) use the formal definition to find the derivative of y = f(x) at x = 1.\n(b) find f(1) and find the equation of the tangent line at the point (1, f(1)).\n(c) graph y = f(x) and the tangent line at the point (1, f(1)) in the same coordinate system.\n(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.\nf(1)=limₕ→0 (1 + h)² + 2-(1)² + 2/h\nevaluate the limit expression to find f(1).\nf(1)=2 (type an integer or a fraction.)\n(b) f(1)=3 (type an integer or a fraction.)\nthe equation of the tangent line at (1, f(1)) is (type your answer in slope - intercept form)

Answer

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)). [ \begin{align*} y-3&=2(x - 1)\ y-3&=2x-2\ y&=2x + 1 \end{align*} ]

Answer:

(y = 2x+1)