MAC 2233 Exam 1 Review Fall 2017

Transcription

1 Fall 2017 This review, produced by the your professor, contains a collection of questions which are represen-tative of the type you may encounter on the eam. Other resources made available by the Teaching Center include: Walk-In tutoring at Broward Hall Private-Appointment, one-on-one tutoring at Broward Hall Walk-In tutoring in LIT 215 Supplemental Instruction Video resources for Math and Science classes at UF Written eam reviews and copies of previous eams The teaching center is located in the basement of Broward Hall: You can learn more about the services offered by the teaching center by visiting

2 1. Solve for : 2( + 1) 1/3 4/3 ( + 1) 2/3 2/3 = Perform the operation and simplify the epression Solve the inequality f( + h) f() 4. Find and simplify h (a) f() = (b) f() = + 4 for 5. Let f() = 2 and g() = (a) Find (f g)() and its domain. (b) Find (g f)() and its domain. 6. Let f() = 1 and g() =. Find f () and its domain. 1 g 7. Sketch the graph of f() = by using a formula to find the verte. Show all intercepts. Confirm your work by writing f in standard form, i.e., f() = a( h) 2 + k. Hint: complete the square and use what you know about transformations and translations. 8. Sketch the graph of f() = 2 = 1 as a transformation of the graph of y =. List each step in transforming the one graph into the other. 9. Use the definition of absolute value to write the function g() = as a piecewise function, then sketch its graph. CLAS Teaching Center 2

3 10. Let f() = 4. Find the inverse function f 1 (), including its domain. Sketch the graph of both functions. 11. Find the inverse of the one-to-one function f() = Use the inverse you found to determine the range of f(). Find, if possible, any asymptotes for f(). 12. Find the domain of the following functions: (a) f() = ( ) 8 (b) f() = ln Find the solution set: (a) log 3 (2 2 5 log 3 () = 1. (b) = ( ) (c) ln( + 8) + ln( 2) = ln(3 + 2). 14. Find the inverse of the function f() = e Sketch the graphs of both f() and f 1 () on the same plane. Include at least one point on the graph of each function and any asymptotes possessed by both. CLAS Teaching Center 3

4 + 4 < Consider the function f() = 2 2 < 2. ln( 1) > 2 (a) Find the following function values if they eist. If they do not eist eplain why (i) f( 4) (ii) f( 2) (iii) f(0) (iv) f(2) (v) f(e + 1) (b) Sketch the graph of y = f(). (c) Use your graph to evaluate each of the following limits. If they do not eist, eplain why. (i) lim f() 2 (ii) lim f() 0 (iii) lim f() Let f() = (a) Find the domain of f(). (b) Find all intercepts (epressed as ordered pairs (, y)). (c) Find all vertical and horizontal asymptotes. (d) Sketch the graph of y = f(), carefully labeling any discontinuities. (e) Use your graph to find lim 2 f(). 17. There is a linear relationship between temperature in degrees Celsius, C, and degrees Fahrenheit, F. Water freezes at 0 C (32 F ) and boils at 100 C (212 F ). Find the function whose domain is degrees Celsius and whose range is degrees Fahrenheit. What is the temperature in degrees Fahrenheit if the temperature is 30 C? What does the slope of the line tell you? 18. The Financial department of a company has produced the following data, where C() is the cost in hundreds of dollars to produce items: C() (a) Use linear regresion to calculate the least squares line and to find the correlation coefficient. (b) According to this model, what are the fied costs of producing this item? (c) What are the anticipated costs of producing 1000 items? (d) What is the marginal cost? What does that tell you? CLAS Teaching Center 4

5 19. The demand and supply functions for a given product are p = D(q) = 60 2q 2 and p = S(q) = q 2 + 9q + 30 where q is the quantity in thousands and p is the unit price. Find the equilibrium quantity and price. 20. A financial manager at Target has made the following observations about a certain product in one of its districts: an average of 250 units will sell in a month when the price is $15, but an average of 50 more will sell if the price is reduced by $1. Assuming the demand function is linear, (a) Epress p as a function of. (b) Find the revenue function, R(), and the production level which will maimize revenue. What is this ma? (c) If fied costs are 800 and the marginal cost is $10 per item, find each value of for which the company will break even. What is the profit at those values? (d) Find the profit function P (). What price should the manager charge to maimize the profit from this item? 21. A farmer plans to spend $6000 to enclose a rectangular field with two kinds of fencing. Two opposite sides will require building materials that cost $3 per foot, while the other two sides can be built using material that costs $2 per foot. (a) Epress the area of the enclosure as a function of the side-length upon which the more epensive fencing must be used. (b) Find the dimensions which maimize the area of the field. 22. Rewrite the epressions as the sum, difference, or multiple of several logarithms. ( ) (a) log ( ) e (b) log +1 ( 2) Mr. Jones invested $2500 at 5.5% interest (compounded continuously). How long will it take his account to grow to 4000 if he does not himself add to the principle? 24. How much money must be invested now at 3 1 % (compounded quarterly) in order to have $ in three years? CLAS Teaching Center 5