Homework Assignment 4 Name : Math140 (Section 2.8, 3.1, 3.2) Section : The homework is worth 10 points. Show all of your work and put a box around your final answer. 1. Use the Intermediate Value Theorem to show that there is a solution of the equation, 23 -4x = 1 in (-1, 0). 2. Find an equation of the tangent line to the curve, f(x) = 1 *2 + 1 at (0, 1) by using the limit definition of the derivative. (Don't use Derivative Rules)3. Answer the following questions. (a) Sketch a graph of f(x) = In? 9| and nd the points c such that f'(c) does not exist. (b) Determine the intervals along the xaxis on which the derivative in the following gure is negative. 3' 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Find the derivative of the following functions by using the limit definition of the derivative. (Don't use Derivative Rules) 4. f (20) = 1-x 2 + x 7. f(2) =V5x-1+15. Answer the following questions. (a) Sketch the graph of f'(x) by using the given graph of y = f(x). f(x (b) The number of bacteria after t hours in a controlled laboratory experiment is n = f(t). What is the meaning of the derivative f'(5)?6. If a rock is thrown upward on the planet Mars with a velocity of 10 m/s, its height (in meters) after t seconds is given by H(t) = 10t 136?. [Use Derivative Rules) (a) Find the velocity at time t. (b) With what velocity will the rock hit the surface? 7. From the above graph of y = at), nd all x-value(s) at which f is not differentiable