Doug Clark The Learning Center 100 Student Success Center learningcenter.missouri.edu Overview

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1 Math 1400 Final Exam Review Saturday, December 9 in Ellis Auditorium 1:00 PM 3:00 PM, Saturday, December 9 Part 1: Derivatives and Applications of Derivatives 3:30 PM 5:30 PM, Saturday, December 9 Part : Further Applications of Derivatives, Integration and Applications Doug Clark The Learning Center 100 Student Success Center ClarkDA@missouri.edu learningcenter.missouri.edu Overview Section 1: Differentiation Rules for Differentiation (1.6) The Product and Quotient Rules (3.1) The Chain Rule (3.) Implicit Differentiation (3.3) Exponential and Logarithmic Derivatives (4.3, 4.5) Section : Applications of Derivatives The Slope of a Curve at a Point (1.) The Derivative as a Rate of Change (1.8) Elasticity of Demand (5.3) First and Second Derivative Rules (.) First and Second Derivative Tests and Graphs (.3) Curve-Sketching (.4) Optimization (.5,.6,.7)

2 Section 1: Differentiation 1. Differentiate the following functions: y x x 7 3 a. 4 y 6x 5 x 7 c. y 4x x 3 x 7 d. y x 5 5x 6 y 9 7x f. f x 1x x 5 e. 7 g. i. y y x 6 x 8 5 1x e 3 j. k. y ln6x 7x 6 h. f ( t) 10 9 y 5x 8e 3x. Suppose that x and y are related by the given equations. Use implicit differentiation to find dy a. dx 6 5 y 5x 7x d dt 6 3. Find t lnt 3 4 t e 4 4 x y y x t Section : Applications of Derivatives 1. Find all the points on the curve horizontal. y x 7 x where the tangent line is. The graph of the function has one relative maximum and one relative minimum point. Plot these two points and check the concavity there. Using only this information, sketch the graph. 1 3 f x x 3x 8x 1 3

3 3. Sketch the following curve, indicating all relative extreme points and inflection points. 3 y x 1x 3 4. A rectangular page is to contain 16 square inches of print. The page is to have a 5-inch margin on the top, the bottom and the sides. Find the dimensions of the page that minimizes the amount of paper used. 5. Let f(t) be the amount of oxygen (in suitable units) in a lake t days after sewage is dumped into the lake, and suppose that f(t) is given approximately by f t t 9 9 increasing the fastest? t. At what time is the oxygen content 6. If the demand equation (in dollars) for a certain commodity is p 40 ln x, where x represents the number of units of the commodity, determine the marginal revenue functions for this commodity, and compute the marginal revenue when x = For the demand equation q = 98 p, find E(p), and determine if the demand is elastic or inelastic (or neither) at p = Currently, 300 people ride a certain commuter train each day and pay $16 for a ticket. The number of people q willing to ride the train at a certain price p is q300 5 p. The railroad would like to increase its revenue. a. Is the demand elastic or inelastic at p = 16? Should the price of a ticket be raised or lowered? 9. A pharmacist wants to establish an optimal inventory policy for a new antibiotic that requires refrigeration in storage. The pharmacist expects to sell 4800 packages of this antibiotic at a steady rate during the next year. She plans to place several orders of the same size spaced equally throughout the year. The ordering cost for each delivery is $1, and the carrying costs, based on the average number of packages in inventory, amount to $8 per year for one package. Use this information to answer the following questions: a. Let x be the order quantity and r the number of orders placed during the year. Find the inventory cost in terms of x and r. Find the constraint function c. Determine the economic order quantity that minimizes the inventory cost and then find the minimum inventory cost.

4 Math 1400 Final Exam Review, Part (Sponsored by the Learning Center) Section (Cont.): Applications Optimization (.5,.6,.7) Exponential Growth and Decay and Compound Interest (5.1, 5.) Section 3: Integration & Applications Antiderivatives (6.1) Definite Integrals and the Fundamental Theorem of Calculus (6.3) Areas in the xy-plane (6.4) Applications of the Definite Integral (6.5)

5 Section (Cont.): Applications 1. There are $88 available to fence in a rectangular garden. The fencing for the side of the garden facing the road costs $18 per foot, and the fencing for the other three sides costs $6 per foot. Consider the problem of finding the dimensions of the garden with the maximum area. a. Determine the objective and constraint functions. Express the quantity to be maximized as a function of x. c. Find the optimal value of x and y.. Until recently, hamburgers at the city sports arena cost $4.0 each. The food concessionaire sold an average of 18,000 hamburgers on game night. When the price was raised to $4.60, hamburger sales dropped off to an average of 14,000 per night. a. Assuming a linear demand curve, find the price of a hamburger that will maximize the nightly hamburger revenue. If the concessionaire had fixed costs of $,500 per night and the variable cost is $0.60 per hamburger, find the price of a hamburger that will maximize the nightly hamburger revenue. 3. Let P(t) be the population (in millions) of a certain city t years after 1990, and suppose that P(t) satisfies the differential equation P 0.01 P t, P 0 7 a. Find the formula for P(t). What was the initial population, that is, the population in 1990? c. What is the growth constant? d. What is the population in 1997? e. Use the differential equation to determine how fast the population is growing when it reaches 9 million people. f. How large is the population when it is growing at the rate of 90,000 people per year? 4. A sample of radioactive material disintegrates from 4 to 3 grams in 00 days. After how many days will just grams remain? 5. Five thousand dollars is deposited into a savings account at 3.5% interest compounded continuously. a. What is the formula for A(t), the balance after t years? What differential equation is satisfied by A(t), the balance after t years? c. How much money will be in the account after years? d. When will the balance reach $7000? e. How fast is the balance growing when it reaches $7000? 6. A company invests $60,000 in a CD that earns 8% compounded continuously. How long will it take for the account to be worth $90,000?

6 7. At what interest rate compounded continuously must money be invested to triple in 4 years? 8. How much must you invest now at 4.% interest compounded continuously in order to have $10,000 at the end of 5 years? 9. Suppose the annual sales S (in dollars) of a company may be approximated t empirically by the formula S 30,000 e where t is the number of years beyond some fixed reference date. Use a logarithmic derivative to determine the percentage rate of growth of sales at t = 5. Section 3: Integration & Applications 1. Find the following indefinite integrals: 4 5 a. 7dx x c. e 7x dx x dx. Evaluate the following definite integrals: 5 a. 5 3x dx 7x ln8 0 e x 8 x e dx f x 3. Find the area of the region between and x = Find the area bounded by f x x and gx x 9 x and the x-axis from x = Find the average value of f x x 13 from x = 0 to x = Find the consumer s surplus for the following demand curve at the given sales level x: 800 p 5 x 16 at x = An investor pays 8% interest compounded continuously. If money is invested steadily at a rate of $0,000 per year, how much time is required until the value of the investment reaches $00,000?

7 8. Find the volume of the solid of revolution generated by revolving about the x-axis the region under the curve y 5 x from x = -5 to x = 5.