1.8. THE TANGENT LINE APPROXIMATION
85
4. For a certain function f , its derivative is known to be f ′ (x) = (x − 1)e −x 2 . Note that
you do not know a formula for y = f (x).
(a) At what x-value(s) is f ′ (x) = 0? Justify your answer algebraically, but include a
graph of f ′ to support your conclusion.
(b) Reasoning graphically, for what intervals of x-values is f ′′ (x) > 0? What does
this tell you about the behavior of the original function f ? Explain.
(c) Assuming that f (2) = −3, estimate the value of f (1.88) by finding and using
the tangent line approximation to f at x = 2. Is your estimate larger or smaller
than the true value of f (1.88)? Justify your answer.
85
4. For a certain function f , its derivative is known to be f ′ (x) = (x − 1)e −x 2 . Note that
you do not know a formula for y = f (x).
(a) At what x-value(s) is f ′ (x) = 0? Justify your answer algebraically, but include a
graph of f ′ to support your conclusion.
(b) Reasoning graphically, for what intervals of x-values is f ′′ (x) > 0? What does
this tell you about the behavior of the original function f ? Explain.
(c) Assuming that f (2) = −3, estimate the value of f (1.88) by finding and using
the tangent line approximation to f at x = 2. Is your estimate larger or smaller
than the true value of f (1.88)? Justify your answer.