Question

4. (-/2 points)

DETAILS
LARAPCALC10 2.5.032.
MY NOTES
ASK YOUR TEACHER
Find an equation of the tangent line to the graph off at the point (2, 2)).
f(x) - 2 (7x - 2)
y-
Use a graphing utility to graph the function and the tangent line in the same viewing window. See Example 4.
у
X
- 2
2.
80 000
20000
60 000
-40000
40 000
-60 000
20000
-80 000
-4
-2
2

Answer 1

To find the equation of the tangent line to the graph of the function f(x) = 2(7x - 2) at the point (2, f(2)), we calculate the derivative to get the slope, evaluate it at x = 2, and use the point-slope form with the point (2, 24) to obtain the equation y = 14x - 4.

Step-by-step explanation:

The student is seeking to find an equation of the tangent line to the graph of a function at a given point. The function is f(x) = 2(7x - 2), and the point of interest is (2, f(2)). To accomplish this, we first need to find the derivative of f(x) which gives us the slope of the tangent line at any point on the curve. After finding the derivative, we evaluate it at x = 2 to find the slope at the point of interest. We then use the point-slope form of a line, which is y - y1 = m(x - x1), where (x1, y1) is the point on the graph, and m is the slope, to find the equation of the tangent line.

1. Calculate the derivative of f(x) to obtain the slope function. Since f(x) = 2(7x - 2), f'(x) = 2(7).
2. The slope of the tangent line at x = 2 is f'(2) = 2(7) = 14.
3. The y-coordinate for the point (2, f(2)) is given by the function f(2), which evaluates to 2(7(2) - 2) = 2(14 - 2) = 24.
4. Therefore, the point is (2, 24).
5. Using the point-slope form, the equation of the tangent line is y - 24 = 14(x - 2) or y = 14x - 4 after simplification.