Which of the following represents the linear approximation to the function f(x)=x3+5x2−9x−5 in the vicinity of the point (1,−8) ?
To find the linear approximation to the function f(x) = x^3 + 5x^2 - 9x - 5 in the vicinity of the point (1,-8), we can use the tangent line equation.
Let's find the slope of the tangent line at x = 1 by taking the derivative of f(x):
f'(x) = 3x^2 + 10x - 9
Substitute x = 1 into f'(x) to find the slope:
f'(1) = 3(1)^2 + 10(1) - 9
f'(1) = 3 + 10 - 9
f'(1) = 4
Now let's find the equation of the tangent line using the slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.
Using the point (1, -8) and the slope m = 4:
-8 = 4(1) + b
-8 = 4 + b
b = -12
Therefore, the equation of the tangent line is y = 4x - 12.
This represents the linear approximation to the function f(x) = x^3 + 5x^2 - 9x - 5 in the vicinity of the point (1,-8).