Question

find an equation of the tangent line to the graph of y = g(x) at x = 6 if g(6) = -3 and g(6) = 4. (enter your answer as an equation in terms of y and x.)

1. - / 2 points

if the tangent line to y = f(x) at (6, 3) passes through the point (0, 2), find f(6) and f’(6).
f(6) =
f’(6) =

Explanation:

First Problem (Equation of Tangent Line to \( y = g(x) \) at \( x = 6 \))

Step 1: 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. For the tangent line to the graph of \( y = g(x) \) at \( x = 6 \), the point on the tangent line (and on the graph of \( g(x) \)) is \( (x_1,y_1)=(6,g(6)) \). We know that \( g(6)=- 3 \), so the point is \( (6,-3) \). The slope of the tangent line at \( x = 6 \) is given by the derivative of \( g(x) \) at \( x = 6 \), i.e., \( m = g^{\prime}(6) \). We know that \( g^{\prime}(6)=4 \).

Step 2: Substitute into the point - slope form

Substitute \( x_1 = 6 \), \( y_1=-3 \), and \( m = 4 \) into the point - slope form \( y - y_1=m(x - x_1) \). We get \( y-(-3)=4(x - 6) \).

Step 3: Simplify the equation

Simplify \( y + 3=4x-24 \). Then, subtract 3 from both sides to get \( y=4x-24 - 3 \), which simplifies to \( y = 4x-27 \).

Step 1: Find \( f(6) \)

The point \( (6,3) \) is on the graph of \( y = f(x) \) (since the tangent line at \( x = 6 \) touches the graph of \( f(x) \) at \( (6,3) \)). By the definition of a function, if \( x = 6 \), then \( f(6) \) is the \( y \) - value of the point on the graph of \( f(x) \) at \( x = 6 \). So \( f(6)=3 \).

Step 2: Find \( f^{\prime}(6) \)

The slope of the tangent line to \( y = f(x) \) at \( x = 6 \) is \( f^{\prime}(6) \). The tangent line passes through the points \( (6,3) \) and \( (0,2) \). The slope \( m \) between two points \( (x_1,y_1)=(6,3) \) and \( (x_2,y_2)=(0,2) \) is given by the formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Substitute \( x_1 = 6 \), \( y_1 = 3 \), \( x_2=0 \), and \( y_2 = 2 \) into the slope formula: \( m=\frac{2 - 3}{0 - 6}=\frac{-1}{-6}=\frac{1}{6} \). Since the slope of the tangent line at \( x = 6 \) is \( f^{\prime}(6) \), we have \( f^{\prime}(6)=\frac{1}{6} \).

Answer:

\( y = 4x-27 \)

Second Problem (Finding \( f(6) \) and \( f^{\prime}(6) \))