Math 1341 Midterm A Fall 2017 PDF

Midterm A – Math 1341, Fall 2017 Show your work for full credit! NAME: 1. (a) (2 √ pts) Find the average rate of change of the function f (x) = 2 − 2x + 3 as x changes from 3 to 11. √ (b) (5...

Midterm A – Math 1341, Fall 2017 Show your work for full credit! NAME: 1. (a) (2 √ pts) Find the average rate of change of the function f (x) = 2 − 2x + 3 as x changes from 3 to 11. √ (b) (5 pts) Compute the derivative of the function f (x) = 2 − 2x + 3 f (a + h) − f (a) at x = 3 using the definition of the derivative f 0 (a) = lim. h→0 h 2. Suppose a particle is moving in the xy-plane, and its position at time t (in seconds) is given by p(t) = (1/(t + 1), 1 + e2t ), where the units in the x and y directions are given in meters. (a) (3 pts) Find the speed of the particle at time t = 1. (c) (2 pts) Find the acceleration of the particle at time t = 1. 1 3. Consider the function f (x) = x4 − 18x2 − 1. (a) (3 pts) On which open intervals is f (x) concave up? On which concave down? (b) (3 pts) Find the critical points of f (x). (c) (2 pts) On which intervals is f (x) is monotone increasing? On which is it monotone decreasing? (d) (2 pts) Find all the local minimum points and all local maximum points of f (x). Explain your answer! 2  Show your work for full credit! NAME: 4. (3 pts each) Differentiate the following functions. Show your steps, not just the final answer. 2 (a) f (x) = (cos x)ex +1 + 2sin x 3 q (b) f (x) = eln(2x−x ) + cos(3x) x2 +4 (c) f (x) = cos x 5. (5 pts) Use logarithmic differentiation (no other method!) to find the derivative of the function x2 · esin x cos x f (x) = (x2 + cos x)4 3 6. Consider the function f (x) = x3 − 2x2 + x. (a) (4 pts) Find the equation of the tangent line at the point on the graph where x = 2. (b) (4 pts) At which other point of the graph is the tangent line parallel to the tangent line at x = 2? 7. Let f (x) = 1/(x+1)+sin x. This function is invertible on the (−1, π/2] interval. (a) (3 pts) Find f −1 (1). (b) (3 pts) Find (f −1 )0 (1) 4