Additional Review Exam 1 MATH 2053 Please note not all questions will be taken off of this. Study homework and in class notes as well!

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1 Additional Review Exam 1 MATH 2053 Please note not all questions will be taken off of this. Study homework and in class notes as well! x Calculate lim x 1 x + 1. (a) 2 (b) 1 (c) (d) 2 (e) the limit does not exist 2. Calculate lim x 2 (x2 x + 4). 3. Calculate lim x 3 x 2 9 x 3. (a) 3 (b) 6 (c) 9 (d) (e) does not exist 1

2 4. The graph of f(x) is given below, use the graph to find lim x 3 f(x). (a) 4 (b) 3 (c) 3.5 (d) 0 (e) does not exist 5. The graph of f(x) = e x is given below, use the graph to find lim x ex. (a) lim x ex = e (b) (c) (d) lim x ex does not exist lim x ex = lim x ex = 0 (e) None of these 6. The graph of f(x) = e x + 2 is given below, use the graph to find lim x f(x) if it exists. (a) the limit does not exist (b) (c) 5 (d) 3 (e) 2 2

3 3 7. Find lim x 2x 1 (a) 3 2 (b) does not exist (c) (d) 0 (e) none of the above 8. Find lim x 2x + 1 3x 4 (a) 2 3 (b) does not exist (c) (d) 0 (e) none of the above 9. Find the instantaneous rate of change of g(t) = 5 t 2 at t = 5. 3

4 10. Let f(x) = 7x answer parts (i) to (iii) (i) Find the average rate of change of the function f(x) = 7x over the interval [5, 7] (ii) Find the average rate of change of the function f(x) = 7x over the interval [ 1, 1] (iii) find the instantaneous rate of change of the function f(x) = 7x at x = 5. 4

5 11. Suppose customers in a hardware store are willing to buy N(p) boxes of nails at p dollars per box, as given by N(p) = 80 5p 2 ; 1 p 4. Find the instantaneous rate of change of demand when the price is $ Use the graph below to estimate the average rate of change of the percentage of new employees from 2000 to

6 13. Suppose the total profit in hundreds of dollars from selling x items is given by P (x) = 2x 2 7x+5. (i) Find the average rate of change as profit changes from 3 to 5. (ii) Find and interpret the instantaneous rate of change of profit with respect to the number of items produced when x = 3. (iii) Find the marginal profit when 5 items are sold. 6

7 14. List the points in the graph in the interval 1 < x < 6 at which the function is not differentiable. 15. For the function shown below answer questions (i) through (iii) (i) What is the interval(s) on which the rate of change is positive? (ii) What is the interval(s) on which the rate of change is negative? (iii) What x values is the rate of change 0? 7

8 16. Suppose the demand for a certain item is given by D(p) = 3p 2 5p where p represents the price of the item in dollars. (i) Find the rate of change of demand with respect to price. (ii) The rate of change of demand when price is $11 is 71. Choose the correct interpretation below. (a) When the price is $11, demand is decreasing at a rate of about 71 items for each increase in price of $11 (b) When the price is $11, demand is increasing at a rate of about 71 items for each increase in price of $11 (c) When the price is $11, demand is increasing at a rate of about 71 items for each increase in price of $1 (d) When the price is $11, demand is decreasing at a rate of about 71 items for each increase in price of $1 17. For f(x) = x 2 + x, find the equation of the tangent line when x = 4. 8

9 18. Let f(x) = x 2 2x, find the instantaneous rate of change using the definition of the derivative. 19. Let f(x) = 3, find the instantaneous rate of change using the definition of the derivative. 5x 20. Explain the relationship between the slope and the derivative of f(x) at x = a. Choose the correct answer below. (a) The derivative of f(x) at x = a describes the rate of change for the slope of the function at x = a. (b) The derivative of f(x) at x = a equals the slope of the function at x = a (c) The slope of the function at x = a describes the rate of change for the derivative of f(x) at x = a (d) The derivative of f(x) at x = a is unrelated to the slope of the function at x = a (e) None of the above. 9

10 21. Explain the concept of marginal cost. How does it relate to cost? How is it found? (i) How does marginal cost relate to cost? (a) Marginal cost refers to the rate of change of cost. (b) Cost refers to the rate of change of marginal cost. (c) Marginal cost is the same as cost. (ii) How is marginal cost found? (a) Marginal cost is found by taking the derivative of cost. (b) Marginal cost is found by taking the anti-derivative of cost. (c) Marginal cost is the same as cost. 22. Assume that a demand equation is given by p = q. Find the marginal revenue for the 100 production level q = 1400 units. 23. Find the derivative of the below functions: (i) f(x) = x 3 12x + x

11 (ii) f(x) = 10x x 7 (iii) f(x) = 5 4 x (iv) f(x) = (5x 2 + 2)(5x 2) (v) g(x) = x2 4x + 1 x

12 (vi) h(x) = (4x2 + 4)(5x + 2) 9x Suppose that f(x) and g(x) are differentiable functions such that f(2) = 1, f (2) = 6, g(2) = 4 and g (2) = 9. Find: (i) h (2) when h(x) = f(x) g(x). (ii) h (2) when h(x) = f(x) g(x). (iii) h (2) when h(x) = (f g)(x). 12

13 25. The total cost (in hundreds of dollars) to produce x units of a product is (i) Find marginal cost function. C(x) = 2x 1 6x + 7. (ii) Find marginal average cost function. (iii) Find marginal average cost for 10 units. (iv) What is the average cost to produce items without bound. (note this answer may not make sense.) 13

14 26. A company that makes bicycles has determined that a new employee can assemble M(d) = 104d2 5d bicycles per day after d days of on-the-job training. (i) Find the rate of change function for the number of bicycles assembled with respect to time. (ii) Interpret M (2). (a) After 2 days of on the job training, the new employee can assemble about 3.7 more bikes than after 1 day of on the job training. (b) After 2 days on the job, the new employee can assemble 3.7 bikes/day. (c) After 2 days of on the job training, the new employee can assemble 3.7 bikes/day. (d) After 2 days on the job, the new employee needs to learn to assemble 3.4 more bikes/day. (e) none of the above 27. Find the derivative of the following functions: (i) g(x) = (x 2 + 3x) 7 (ii) f(x) = 47(9x 3 8) 3/2 14

15 (iii) h(x) = (5x + 1) 5 (4x + 1) Consider the following table of values of the functions f and g and their derivatives at various points. (i) Find d [g(x) + f(x)] at x = 4 dx (ii) Find d [g(x) f(x)] at x = 1 dx (iii) Find d dx ( ) g(x) at x = 2 f(x) (iv) Find d [(g f)(x)] at x = 3 dx 15

16 29. Find the equation of the tangent line to the graph of at x = 5. f(x) = x