1 This set of problems is for preparing for the final exam of Math 190. You need to do more problems in order to perform well on the test. 1. Use a graph to find a number δ such that sin x 1 0.2 whenever x . 6 2 2. If 1 f ( x) x 2 8x 8 for all x, find lim f ( x) . x1 3. (a) Use the definition of derivative to find the slope of the tangent line to the graph of the exponential function y 8 x at the point (0,1). (b) Estimate the slope to three decimal places. 4. Use the midpoint rule with n = 4 to approximate the area of the region bounded by the given curves. y 1 x 3 , y 1 2 x , x 2 . 5. Find the area of the region bounded by the parabola y x 2 , the tangent line to this parabola at (6,36), and the x-axis. 5 6. Evaluate the integral. 4x x 2 dx 2 x t2 1 1 7. If F ( x) f (t ) dt , where f (t ) 10 u 4 du , find F " (2). u 8. Find the inflection point for the function. f ( x) 6 5 x 2 sin x, 0 x 3 . 9. Use Newton’s method with the initial approximation x1 =2 to find x 4 , the fourth approximation to the root of the equation x 4 13 0 .(Give your answer to four decimal places.) 10. Consider the figure below, where AC = 7 ft, BD = 1ft and AB = 6 ft. How far from the point A should the point P be chosen on the line segment AB so as to maximize the angle θ? Round you answer to the nearest hundredth. 2 11. Evaluate the limit. lim 7 x 1 7 1 x x 0 12. Use implicit differentiation to find an equation of the tangent line to the curve 4 x 2 3 y 2 7 at the point (1,1). 13. Find the limit. lim 1 10 x 1/ x x0 14. Find the exact values of the numbers c that satisfy the conclusion of The Mean Value Theorem for the function f ( x) x 3 5 x for the interval [-5,5]. 1 15. Find tan( 2 ) d . 0 16. The velocity of a particle is v(t ) t 3 10t 2 24t ft / s . Compute the (a) displacement over [0,4], [4,6], and [0,6]. (b) Total distance traveled over [0,6]. 17. Find lim x1 1 x2 . cos 1 x 18. It costs you c dollars each to manufacture and distribute backpacks. If the backpacks sell at x dollars each, the number sold is given by a n b(100 x) where a and b are positive constants. What selling price xc will bring a maximum profit?