Find an equation of the tangent line to y = f(x) at x = 2.
A) y = 9x - 16
B) y = -24x - 27
C) y = 24x - 27
D) y = 24x + 27

Question

Find an equation of the tangent line to y = f(x) at x = 2. 
A) y = 9x - 16 
B) y = -24x - 27 
C) y = 24x - 27 
D) y = 24x + 27

Quizplus · Accepted Answer

To find the equation of the tangent line to y = f(x) at x = 2, we need to find the slope of the tangent line (which is the derivative of f(x) at x=2) and the y-intercept of the tangent line.

Given the function y = f(x), we don't have enough information to directly determine f(x) or its derivative. Therefore, we can use the limit definition of the derivative to find the slope of the tangent line:

f'(2) = lim(h -> 0) [f(2 + h) - f(2)] / h

Without further information about f(x), we cannot simplify this expression. However, we can use the fact that we are given answer choices to estimate the value of f'(2):

A) y = 9x - 16
The slope of this line is 9, which cannot be the slope of the tangent line to any function at x = 2, since the tangent line must "touch" the curve at that point.

B) y = -24x - 27
The slope of this line is -24, which is also unlikely to be the slope of the tangent line to a smooth function at x = 2.

C) y = 24x - 27
The slope of this line is 24, which is a reasonable estimate for f'(2) for some functions.

D) y = 24x + 27
The slope is the same as in choice C, but the y-intercept is different. Therefore, this line is parallel to the line in choice C but does not intersect the function at x = 2.

Based on these estimates, we can choose C as the best option for the equation of the tangent line. Note that this method is not foolproof and may not always lead to the correct answer, so it should be used with caution.

Find an Equation of the Tangent Line to Y =