1a) Find the tangent line approximation to f (x) = ex near x = 0.
b) If you use your tangent line to estimate values of f , will your estimates be larger or smaller than the true value? 2) Consider the function p(x) = x5 − 5x4 + 10x3 − 10x2 + 2x + 3.
a) One can check that p(1) = 1 and p(2) = −1. Since p(x) is a continuous function, the Intermediate Value
Theorem tells us that the following equation x5 − 5x4 + 10x3 − 10x2 + 2x + 3 = 0 has a solution for some x value between 1 and 2. Find an approximate value for this solution by replacing the left side of the above equation by its linearization near x = 1.
b) Check that your answer from part a satisfies (approximately) the equation p(x) = 0.
3) A runner runs a 100 meter dash, and their distance (in meters) as a function of time (in seconds) is given by s(t) = t2 . Since s(10) = 102 = 100, we see that the runner completes the dash in exactly 10 seconds.
(In case your curious, Wikipedia says that the current men’s world record is 9.58 seconds set by Usain Bolt, and the current women’s world record is 10.49 seconds set by Florence Griffith-Joyner.)
a) Compute the runner’s average velocity for the entire dash.
b) Since s(t) is continuous and differentiable, the Mean Value Theorem says that at some time during the dash, the runner’s instantaneous velocity is equal to the average velocity from part a. Find this time.
4) Find all local maxima and minima of the function f (x) = 3x4 + 8x3 − 17.
5) Find the inflection points of g(x) =