1. Use the Intermediate Value Theorem to show that there is a solution of the equation, x3-4x = 1 in (-1, 0). 2. Find an equation of the tangent line to the curve, f(x) = - 1 x2 + 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) = |x2 - 9| and find the points c such that f'(c) does not exist. (b) Determine the intervals along the r-axis on which the derivative in the following figure is negative. 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 X 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Find the derivative of the following functions by using the limit definition of the derivative. (Don't use Derivative Rules) 4. f(2) = 1 - x 2+ x 7. f(x) = V5x -1+15. Answer the following questions. (a) Sketch the graph of f'(x) by using the given graph of y = f(x). If ( > (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 HOE) = 10: 1.86152. (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 = f(x), nd all mvalue(s) at which f is not di'erentiable