EXTRA (take home) 1. Differentiability Let g(x) be the piecewise function defined below. Find the values of a and b that make g(x) differentiable at x=1. 9(x) = *2 - 3x+3, x 51 lax + b, x > 1 2. Intermediate Value Theorem a. State the Intermediate Value Theorem. (Be sure to include the assumptions.) b. Apply the Intermediate value theorem to verify the equation x3 - 3x2 +2 = 0 has a solution between x = 2 and x = 3. (Be sure to verify the assumptions are met. Also state clearly the conclusion. 3. Logarithmic Differentiation. Find dy/dx for each of the following by applying logarithmic differentiation y = sin(arccos x) y = arctan x - 1+ x 2 4. Derivative of Inverse Consider the function f(x) = 2x3 + 4x - 1 a. Show P(x) > 0 for all real valued x. Why does this imply an inverse exists? b. Apply the theorem regarding derivatives of inverses to find [f -1]' (5) (the derivative of the inverse f-1(x) at x = 5.) 5. Derivative of Trigonometric Inverses x3ex2 y = xVx Vx 2+1 6. Related Rates An airplane is flying at an altitude of 5 miles and passes directly over a radar antenna (see figure). When the plane is 13 miles away (s=13), the radar detects that the distance s is changing at a rate of 315 mph. What is the speed of the plane (dx/dt)? X 5 mil S