Practice Problems **Note this list of problems is by no means complete and to focus solely on these problems would be unwise.**

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1 Topics for the Final Eam MATC 100 You will be allowed to use our MATC 100 calculator. The final eam is cumulative (Sections.-., Sections , Sections.1-.5) - see the details below. Sections.-. & : Limits & Differentiation The main theme of chapter is the notion of a limit and how it applies to the definition of the derivative. Chapter 3 introduces ou to the rules (or shortcuts) for finding derivatives. All of these topics were on previous eams. You should epect to be asked some questions regarding calculation of limits graphicall and algebraicall. I guarantee to ask ou to use the limit definition of the derivative to compute a derivative. I guarantee I will ask ou to find the equation of a tangent line to a given function. I will ask ou to find the derivative of something like f 3 7. Sections 3. &.1-.3: Applications of Differentiation up to optimization These sections cover marginal analsis, the first derivative, the second derivative, and asmptotes. Section. - Finding Absolute Etrema Know the difference between absolute and relative (or local) etrema (maimums and minimums). Be able to determine them graphicall. Note that when we refer to the "maimum" (or minimum) that refers to the value. Know the procedure for finding absolute etrema of a continuous function f on a, b. Know the procedure for finding absolute etrema (either the ma or the min) of a continuous function f on an open interval when there is onl one relevant critical number. Want more practice?. Self Check Eercises 1-3 AND Chapter Review Eercises 3-3 Section.5 - Optimization Applications Be able to do optimization problems. You should be able to determine the function to be optimized, the relevant interval, and then appl calculus to find and justif ou have found the absolute etreme value. Want more practice?.5 Self Check Eercises 1-3 AND Chapter Review Eercises 33, 39, 0,, 5, 7, 8 Practice Problems **Note this list of problems is b no means complete and to focus solel on these problems would be unwise.** 1. B hand find the derivative for each of the following. You do not need to simplif. a. f b. f c. f d. f

2 e. f f. f 3 9 g. f Find the second derivative of f Find f 3 if f 3. Recall that a function f is continuous at a if all of the following conditions are satisfied: i. f a is defined. ii. lim iii. a f eists lim a f f a On each set of aes below, sketch the graph of a function that meets the given criteria. a. f is not continuous at because it does not satisf condition (i). b. g is not continuous at 1 because it does not satisf condition (ii) but it does satisf condition (i). c. h is not continuous at because it does not satisf condition (iii) but it does satisf conditions (i) & (ii). d. k is continuous on, but k does not eist a) f is not continuous at b) g is not continuous at c) h is not continuous at d) k is continuous, k DNE e. Consider the graph of f given below.

3 lim f lim lim f lim lim f f 8 f f 8 lim f 8 f 3 3 f 10 f f 8 f 10 f. (5 points) Find the given interval. Write our answer in interval notation. Write "NONE" if appropriate. i. The intervals wheref 0. ii. The intervals where f 0. iii. The intervals where f 0. iv. The intervals where f Consider the function f 5 a. Write down the difference quotient for f in unsimplified form. b. Find the simplified difference quotient for f. c. Find the limit as h approaches zero of the simplified difference quotient. d. What is the name of the function ou found in part b)?. Consider the function f a. Find all the critical numbers of the function. b. How can ou be sure ou found all of the critical numbers? c. Find the intervals over which the function is increasing and decreasing. Give our answer in interval notation. d. Find the relative etreme values for f, if an. Be sure to use calculus to justif our answers. e. Find the absolute etreme values for f, if an. f. Find the intervals over which the function is concave up and concave down. Give our answer in interval notation. g. Find the point(s) of inflection. Be sure to give both the and the coordinates. Eplain what each inflection point tells ou about f. h. What is the absolute maimum and minimum value of the function on the closed interval,? 7. Consider a function f with the following characteristics: f 0 on the interval, 3, 8 1, f 0 on the interval, 3 8, 1 f is undefined, f 8 is undefined 3

4 f 3 0, f 1 0 f, f 3, f 8 is undefined, f 1 f 0 f 0 on the interval,, 8 8, f 0 on the interval, These characteristics can be summarized in "sign charts" as given below.,, 3 3 3, 8 8 8, 1 1 1, f DNE 0 DNE 0 f DNE,, 8 8 8,, f DNE DNE 0 f DNE 11 a. Sketch a possible graph of f on the aes below b. What are the critical numbers for f? c. What are the relative maimum and relative minimum values of f? d. What are the inflection points for f? 8. Consider the function whose graph is shown. f

5 a. Find the equation of the tangent line at 0.. b. Sketch the graph of the tangent line at 0. on the same set of aes above. 9. Find the absolute maimum and minimum values of the function f on the interval,. 10. Find the absolute minimum and maimum values (if the eist) of the function g 1 on 0,. 11. The marketing department of Camcon has determined that the weekl demand for their digital cameras is given b p d where p denotes the camera s unit price (in dollars) and denotes the quantit demanded. a. Find the revenue function R. b. Find the actual revenue gained on the 001st camera. c. Find the marginal revenue when 000. Eplain wh this is so close to our answer in part b). d. Suppose that the marketing department has determined that the weekl cost incurred for producing each camera is $ The weekl fied costs incurred is $ Find a function,c, which represents weekl total cost. e. Find the weekl profit function, P. f. How man cameras should be produced in order to maimize profit? Be sure to use calculus to justif our answer. g. What price should Camcom charge for each camera in order to maimize profit? h. How man cameras should be produced in order to break-even? i. Sketch a graph of the profit which will help the eecutives at Camcom to analze their business practices. Label at least three points on each ais. Show the eecutives where their maimum profit and break-even points occur. 5

6 1. Phillip, the proprietor of a vineard, estimates that the first 10, 000 bottles of wine produced this season will fetch a profit of $5 per bottle. However, the profit from each bottle beond 10, 000 drops b $ for each additional bottle sold. Assuming at least 10, 000 bottles of wine are produced and sold, how man bottles should be sold in order to maimize profit? What will be the maimum profit? Be sure to justif our answer using calculus. 13. A poster is to have an area of 180 in with 1-inch margins at the bottom and sides and a -inch margin at the top. What dimensions will give the largest printed area? What will be the largest area? Be sure to justif our answer using calculus. 1. A rectangular storage container with an open top is to have a volume of 10 m 3. The length of its base is twice the width. Material for the base costs $10 per square meter and material for the sides costs $ per square meter. Find the dimensions of the container that will minimize the cost of materials. What will be the minimum cost? Be sure to justif our answer using calculus. 15. Postal regulations specif that a parcel sent b parcel post ma have a combined length, depth and height of no more than 108 in. In other words, the length plus the depth plus the height must be no more than 108 in. Suppose a rectangular package with equal depth and height has a combined length, depth and height of eactl 108 in. Find the dimensions of the bo that will ield the maimum volume. What will be the maimum volume? 1. A ship carring math students has the misfortune to be wrecked on Calculus Island. The population of the island after time t in ears is given b P t 3. 5t t 3t t 3. 3 a. How man students were initiall shipwrecked? How man students were on the island after

7 1 ear? 3 ears? 10 ears? b. What is the limiting population to the island? That is, what will the population be in the long run? c. At what rate is the population increasing after time t ears? d. How fast is the population increasing initiall? after 1 ear? 3 ears? 10 ears? e. How fast will the population be increasing in the long run? f. Over what interval is the population increasing? Eplain wh this is the same as solving P t 0. g. Over what interval is the population decreasing? Eplain wh this is the same as solving P t 0 h. Over what interval is the rate of population increase increasing? Eplain wh this is the same as solving P t 0. i. Over what interval is the rate of population increase decreasing? Eplain wh this is the same as solving P t The concentration of a certain drug in an organ at an time t (in hours) is given b g t t 1.0 where g t is measured in grams/cubic centimeter (g/cm 3 ). a. What is the initial concentration of the drug in the organ? b. What is the concentration of the drug in the organ after hours? c. What will be the concentration of the drug in the organ in the long run? d. How fast is the concentration of the drug in the organ changing after hours? Be sure to include the units. 7