MATH 201 - Brief Calculus Review for Test 1 1. Estimate the slope of the following function at x = 1. Show (on the graph) how you arrive at your estimate. 2. Use the limit definition to find the derivative of each function. a) b) f ( x) 3x x 2 f ( x) 1 x 3. Describe the x-values for which f is differentiable. Explain your reasoning. a) b) y x3 y x 1 4. Find the derivatives. Simplify your answers. a) b) f ( x) 5 d) y 2x 4 g) g ( x) e) f (t ) t 6 3t 4 t 3 7t 1 y x 1 3x 5 2 3 4 2x 1 2 j) y x2 3 h) 4 x3 5 x2 2 y x 4 c) y f) 5 x6 f (t ) 2t 3 7 i) y 3x 7 2x 1 5 5. Find the slope of the graph of f ( x) x 2 5x function) 6. Find the equation of the line tangent to 7. a) Find the average rate of change of at the point (1,-4). (It is not necessary to graph the x 5 f ( x) x g (t ) t 2 b) Find the instantaneous rate of change of 3 at the point (2, -3/2) over [-1, 3] g (t ) t 2 3 at x = 1 8. 9. The cost, C, and revenue, R,(both in dollars) for producing x units is given as R x 8 x 200 x 3 2 . Find each of the following. a) The marginal cost of producing x units. b) The marginal revenue of producing x units. c) The marginal profit of producing x units. d) The marginal profit of producing 10 units. e) Should the 11th item be produced? Why or why not? 10. C x 2 2000 and 11. Find the point(s), if any, at which the graph of x2 f ( x) x 8 would have a horizontal tangent line. 12. Use the given functions to answer the questions. f ( x ) 4 x 5 3 x 3 2 x 2 7 a) Find c) Find , 3 g ( x) x , b) Find f (x ) d) Find f (2) h( x ) x 5 g (x) h (7) 13. where t is measured in seconds. a) Write the velocity of the rock, v, as a function of the time t. b) What is the acceleration of the rock? 14. Use implicit differentiation to find dy dx for each of the following equations. a) b) x 3 y 2 9 c) x 2 xy y 2 5 15. Write the equation of the line tangent to the graph of 2 2 x y 9 at the point 16. Assume that x and y are both differentiable functions of t. Given that find dx dt x 3 y 1 y2 2 y 4 x 2 2, 1 and dy 3 dt , when x = 1 and y = -2. 17. 18. Answers: 1) about 2. Draw a line tangent to the curve at x=1. Use two points on that line to find the slope. 2a) 6x - 1 b) 1 x2 Be sure to use the definition in section 2.1, NOT the rules from section 2.2 3a) f is differentiable for every x-value except x = -3. The graph has a sharp turn (node) there. b) f is differentiable on 1, , everywhere in the domain except x = 1. The graph has a vertical tangent line there. 4a) 0 e) b) f) 2 3x 5 9 x 5 x 6 3 h) c) 6t 5 12t 3 3t 2 7 2 i) 2 8x 5 2 x 30t 2t 7 2 j) 17 2 x 1 3 g) 4 x 2 d) 30 x7 x 3 2 8 13 x 3 16 x 2x 2 1 2 or x x2 3 x2 3 5) -3 6) y 5 x4 4 7a) 31 b) 27 8a) -80 ft/sec 9a) 2x b) -64 ft/sec and -96 ft/sec b) c) 3x 2 16 x 200 c) 5.89 sec 3 x 2 14 x 200 d) -188.47 ft/sec d) $40 per item e) Yes, the marginal profit at x=10 is 40. That means we can expect an additional $40 in profit if we make the 11th item. 10) P 1 2 x 17.8 x 85000 3000 11) (0, 0) and (16, 32) 12a) 240 x 2 18 b) 6 x3 c) 978 d) 2 0.0884 16 13a) V = -27/5 t + 27 (V is in ft/sec, t is in sec) b) 14a) (multiply both sides of equation by y2 first) 15. 3 x 2y 2 b) y 2 2 x 9 18a) $112.50 per week c) y 2x x 2y 1 9y 2y 27 ft / sec 2 5 2 16. -3 b) $7500 per week 17. $137,500 per week c) $7387.50 per week