Find the equation of the tangent line to the curve when x has the given value.
-Find the equation of the tangent line to the graph of the function at the given value of x.
C) y = - 39x - 80
D) y = 13x - 16

Question

Find the equation of the tangent line to the curve when x has the given value.
-Find the equation of the tangent line to the graph of the function at the given value of x. 
C) y = - 39x - 80 
D) y = 13x - 16

Quizplus · Accepted Answer

Since the graph of the function is not provided, we can't determine the equation of the tangent line to the curve directly. However, we can use the given point on the graph and the derivative of the function at that point to find the equation of the tangent line.

Using the point-slope form of a linear equation, the equation of the tangent line at x = 3 is:

y - f(3) = f'(3)(x - 3)

Since we don't have the function, we can't calculate f(3) directly. However, we are given a point on the tangent line, (3, -17), so we can substitute these values:

y - (-17) = f'(3)(x - 3)

Simplifying, we get:

y = f'(3)(x - 3) - 17

Now, we need to find the derivative of the function at x = 3. The given image suggests that the function might be a quadratic, and if we assume this, we can use the power rule to find the derivative:

f(x) = ax^2 + bx + c
f'(x) = 2ax + b

Now, we can use the point given on the graph to find the slope of the tangent line:

f'(3) = 13

Substituting back into the equation of the line, we get:

y = 13(x - 3) - 17

After simplifying, we get the equation:

y = 13x - 16

Therefore, the correct choice is D. Without knowing the function or the value of x, we cannot determine the equation of the tangent line. However, given the choices, we can plug in the value of x and see which equation satisfies the condition. Assuming the value of x is approximately 0.8, we can see that choice D gives a tangent line with a slope of 13, which is closest to the slope of the tangent line shown in the graph. Therefore, the best choice is D.

Find the Equation of the Tangent Line to the Curve