Chapter 9 Study Guide
Chapter 9 Study Guide Math 1100
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This 5 page Study Guide was uploaded by Shannon L on Monday February 1, 2016. The Study Guide belongs to Math 1100 at University of Utah taught by Bin Xu in Winter 2016. Since its upload, it has received 61 views. For similar materials see Business Calculus in Math at University of Utah.
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Date Created: 02/01/16
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Question from 9.2: Continuous Functions; Limits at Infinity ▯ If an annuity makes an infinite series of equal payments at the end of the interest periods, it is called a perpetuity. If a lump sum investment of A is nneded to result in n periodic payments of R when the interest rate per period is i, then −n An = R 1 − (1 + i) . i (a) Evaluate lim A to find a formula for the lump sum payment for a perpetuity. n → ∞ n (b) Find the lump sum investment needed to make payments of $110 per month in perpetuity if interest is 6%, compounded monthly. (Round your answer to the nearest cent.) $ 4. Question from 9.2: Continuous Functions; Limits at Infinity ▯ Determine whether the function is continuous or discontinuous at the given xvalue. Examine the three conditions in the definition of continuity. 2 f(x) = x + 4 if x ≤ 1, x = 1 2 if x > 1 8x − 2 The function is continuous at x = 1. The function is discontinuous at x = 1. 5. Question from 9.3: Rates of Change and Derivatives ▯ If an object is thrown upward at 128 feet per second from a height of 60 feet, its height S after t seconds is given by the following equation. S(t) = 60 + 128t − 16t 2 (a) What is the average velocity in the first 4 seconds after it is thrown? ft/sec (b) What is the average velocity in the next 4 seconds? ft/sec 6. Question from 9.3: Rates of Change and Derivatives ▯ 2 We are given f(x) = 3x and fௗ'(x) = 6x. (a) Find the instantaneous rate of change of f(x) at x = 4. (b) Find the slope of the tangent to the graph of y = f(x) at x = 4. (c) Find the point on the graph of y = f(x) at x = 4. (x, y) = 7. Question from 9.4: Derivative Formulas ▯ Find the derivative of the function. 5 3 h(x) = 5 − 3 + 8 x x x h'(x) = 8. Question from 9.4: Derivative Formulas ▯ Find the derivative of the function. h(x) = 14x 12 + 6x − 2x + 19x − 7 h'(x) = 9. Question from 9.5: The Product Rule and the Quotient Rule ▯ A travel agency will plan a group tour for groups of size 35 or larger. If the group contains exactly 35 people, the cost is $270 per person. If each person's cost is reduced by $10 for each additional person above the 35, then the revenue is given by the equation shown below, where x is the number of additional people above 35. R(x) = (35 + x)(270 − 10x) Find the marginal revenue if the group contains 40 people. $ Interpret your result. The revenue will decrease by the absolute value of the marginal revenue in dollars if the group adds one person. The revenue will increase by the absolute value of the marginal revenue in dollars if the group adds one person. The revenue will decrease by the absolute value of the marginal revenue in dollars if the group removes one person. The revenue will increase by $1 if the group adds the absolute value of the marginal revenue number of people. The revenue will decrease by $1 if the group adds the absolute value of the marginal revenue number of people. 10. Question from 9.5: The Product Rule and the Quotient Rule ▯ For the function y = (x + 1)(x − 4x), at (−2, 0) find the following. (a) the slope of the tangent line (b) the instantaneous rate of change of the function 11. Question from 9.6: The Chain Rule and the Power Rule ▯ Ballistics experts are able to identify the weapon that fired a certain bullet by studying the markings on the bullet. Tests are conducted by firing into a bale of paper. If the distance s, in inches, that the bullet travels into the paper is given by s = 125 − (5 − 10t) 3 for 0 ≤ t ≤ 0.5 second, where t is the time in seconds after the bullet hits the paper, find the velocity of the bullet onetenth of a second after it hits the paper. ft/sec 12. Question from 9.6: The Chain Rule and the Power Rule ▯ Differentiate the function. f(x) = (2x − 1) 24 fௗ'(x) = 13. Question from 9.7: Using Derivative Formulas ▯ The total physical output P of workers is a function of the number of workers, x. The function P = f(x) is called the physical productivity function. Suppose that the physical productivity of x construction workers is given by P = 7(3x + 6) − 15. Find the marginal physical productivity, dP/dx. dP/dx = 14. Question from 9.7: Using Derivative Formulas ▯ Find the derivative of the function. Simplify and express the answer using positive exponents only. y = (x − 3)(x + 6) y' = 15. Question from 9.8: HigherOrder Derivatives ▯ A particle travels as a function of time according to the formula 3 s = 140 + 9t + 0.09t where s is in meters and t is in seconds. Find the acceleration of the particle when t = 3. m/sec 2 16. Question from 9.8: HigherOrder Derivatives ▯ Find the second derivative. y = x − x y'' = 17. Question from 9.9: Applications: Marginals and Derivatives ▯ The price of a product in a competitive market is $500. If the cost per unit of producing the product is 110 + 0.1x dollars, where x is the number of units produced per month, how many units should the firm produce and sell to maximize its profit? units 18. Question from 9.9: Applications: Marginals and Derivatives ▯ Total revenue is in dollars and x is the number of units. Suppose that in a monopoly market, the demand function for a product is given by p = 420 − 0.2x where x is the number of units and p is the price in dollars. (a) Find the total revenue from the sale of 500 units. $ (b) Find the marginal revenue MR at 500 units. MR = $ Interpret this value. The 501st unit will bring in |MR| hundred dollars more in revenue. The 501st unit will lose |MR| hundred dollars more in revenue. The 501st unit will lose |MR| dollars more in revenue. The 501st unit will bring in |MR| dollars more in revenue. (c) Is more revenue expected from the 501st unit sold or from the 701st? Explain. The 701st unit will bring in $ more in revenue. Thus the ▯▯▯6HOHFW▯▯▯ unit will bring in more revenue.
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