Using Linear Approximation Determine the following for 10 points and 5 stars?

Question

if f(x)= -5x^3-5x+4 then f(1)= -6

The tangent line approximation to the function f(x) at x=1 is L(x)=____________

Determine a value of x near 1 such that L(x)= -12.00 ___________

Use the tangent line approximation to estimate the value of f(1.15) __________________

Answer 1

f'(x) = -15x^2 - 5, so f'(1) = -20, and thus
L(x) = -6 + (-20)(x - 1) = -20x + 14
Next, solve -12.00 = -20x + 14. the value of x is 1.3.
Finally, evaluate L(1.15): -9

Answer 2

slope of the tangent line is -15x² - 5, at x = 1 that's -20. So the tangent line through (1,-6) is y+6 = -20(x-1), or L(x) = -20x + 14.

if L(x) = -12,
-12 = -20x + 14
-26 = -20x
x = 1.3

f(1.15) ≈ f(1) + f'(1)•(0.15) = -6 + -20(0.15) = -9

Answer 3

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