MATH 122A FINAL EXAM STUDY GUIDE (Spring 2014)

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1 MATH A FINAL EXAM STUDY GUIDE (Spring 0) The final eam for spring 0 will have a multiple choice format. This will allow us to offer the final eam as late in the course as possible, giving more in-class time to cover material. More information about the final eam, along with procedures for taking the eam will be posted in DL later in the course. The format of the questions in this study guide is not multiple-choice in order to encourage you to solve the problems completely. These are not samples of questions that will appear on the final, but they do provide practice for the material that will be covered. Answers will be provided in a separate file.. Find the domain of the following functions: gt ( ) t 0 = b) hy ( ) = y + 5 y. Gasoline is being pumped into a tank at a constant rate (cubic feet per minute). For each of the tanks below, sketch a graph of the height of the water in the tank as a function of time. You can assume the tank is initially empty and will be filled. Also assume the shape and orientation of the tank is as shown. b) c). The lift L on an airplane wing at take-off is proportional to the square of the speed s of the plane and the area A of its wings. Write an equation for lift. b) If the speed is only half as much, how much larger should the area of the wings be for the lift to be the same?. Suppose the target heart rate when eercising, R, is a function of a person s age, A, and is given by the linear formula R H0 = p(0 A H0). Assume H 0 and p are constants. Find the slope and give a practical interpretation.

2 5. For certain category hurricanes, the cube of the diameter D (in miles) of the hurricane is roughly proportional to the square of the hurricane s duration t (in hours). Write an equation to represent the relationship between diameter of the hurricane and duration. Solve your equation for D. b) If the diameter of hurricane Veronica was 0 miles with a duration of 7 hours, what would be the diameter of a hurricane with a duration of 0 hours according to this model? 6. From the given data, determine if Account Balance is a function of Week. Account Balance $,50 $,500 $,50 $,700 $,500 $,800 Week Let 5 pt () = t +. Find pt ( + h) pt () h and simplify as much as possible. 8. Suppose an oil spill covers a circular area and that the radius increases according to t rt () = + where t represents the number of minutes since the spill was first observed. The radius is measured in inches. What was the radius of the spill when it was first observed? b) Epress the area A of the oil spill as a function of t. c) Find the eact time when the area of the spill is 8π square inches. 9. Use the graph of g ( ) given at the right to answer the questions. If f ( ) b) Find the range of = +, find g( f ()). g ( ). c) Is y= g ( ) a one-to-one function? g ( ) g() g() d) Find. What does this number represent on the graph of g? ( ) e) For what value(s), if any, would y = be undefined on the interval [ 0,5 ]? g ( ) 0 f) Does gappear ( ) to be concave up or concave down on the interval.5.5?

3 0. For functions e f () t + f t () 5t = and gt () = t+, find and simplify: b) ( ) f() t + gt () c) gt ( ) d) f( t ). One of the functions below is linear, one is eponential, and one is quadratic. Determine which is which and then find a formula for each. Table A Table B Table C f( ) g ( ) ( ) h Sketch a graph of y = f( ) that satisfies all of the following conditions: * domain < < * f( ) is an odd function * range < y < + * f( ) is not invertible. Let f ( ) = A + B and g( ) = C + D where A 0, C 0. Find What are the slope and vertical intercept of h ( )? h ( ) = ( g f) ( ).. If the zeros of gare ( ) = and =, what are the zeros of y= g ( + )? 5. Let f( ) = and g ( ) =. Find ( ( )) f g and simplify your answer. 6. Let (, 5) be a point on the graph of y = f( ). Find the corresponding point on the graphs of each of these transformations. y = f( ) + b) y = f c) y = f( ) 7. It is predicted that the population of a particular state will double every 5 years. Determine the annual and monthly growth rates. Epress your answers as percents. b) Determine the continuous growth rate per year. Epress your answer as a percent.

4 8. A typical cup of coffee contains about 00 mg of caffeine. Every hour approimately 6% of the amount of caffeine in the body is metabolized and eliminated (decays). Write an equation for the amount of caffeine, C, in the body as a function of t, the number of hours since a single cup of coffee was consumed. b) Find the time when 0% of the caffeine has been metabolized and eliminated from the body. Give both an eact answer and a decimal answer. Include units. 9. In 997, the average tuition at four-year public universities was $,60 per year. In 998, that figure rose to $,0 per year. If tuition increased linearly, write a formula for the tuition as a function of years since 997. Use your function to estimate the tuition in 006. b) If tuition increased eponentially, write a formula for the tuition as a function of years since 997. Use your function to estimate the tuition in 006. Note: The average tuition in 006 was actually $,0. 0. Let f( ) = log (8 7 ) Find the domain and range. b) Use algebra to find the intercepts of f( ). Simplify your answers.. Fish are introduced into a large lake system. The population size (in numbers of fish) can be 0.0t modeled by Pt ( ) = e where t is measured in months since the fish were introduced. Find P () and give a practical interpretation. b) Find P (500) and give a practical interpretation. c) Is the population of the fish increasing or decreasing? d) When does the population size reach 000 fish according to the model? e) What happens to the population size as t?

5 . Epand the following epression completely and simplify: b) Combine the following epression into a single logarithm: ln( + ) ln( ) + ln( + ) log a y a 5. The graph of y = ln along with a line passing through intersecting points A and B is shown below. Find the coordinates of A and B. B b) Find the equation of the line. A e. Answer true or false: The domain of all polynomials is (, ). b) If gpasses ( ) the vertical line test, it is a one-to-one function. c) An even degree polynomial must have at least one maimum or minimum. d) All rational functions have vertical asymptotes. e) If y = f( ) is an odd function, then y = f( ) is an even function. 5. Answer the following questions about the polynomial graphed below. What is the smallest possible degree? b) Is the leading coefficient positive or negative? c) Write a possible equation for this polynomial.

6 6. A cable must be laid from point A to a point C across a river. The plan is to go from point A to point B under water and then continue from point B to point C on land. The cost of cable laid under water is $7 per foot while the cable laid on land is $8 per foot. Write an equation for the total cost of laying the cable in terms of d. 7. Find all intercepts, asymptotes, and holes (if any) for + 5 f( ) =. ( + 5)(5 ) 8. In each case, find the value(s) of k so that the following is true for t + k pt () = t +. p () = 5 b) p () = 0 c) The graph of pt () has no zeros. 9. Each time a person s heart beats, their blood pressure increases and then decreases as the heart rests between beats. A certain person s blood pressure is modeled by the function b( t) = A + Bsin( Ct) where bt () is measured in mmhg and t is measured in minutes. Find values for A, B, and C if the person s average blood pressure is 5 mmhg, the range in blood pressure is 50 mmhg, and one cycle is completed every /80 of a minute. 0. The Columbia Tower in Seattle is 95 feet tall. The Seafirst Tower is T feet tall and stands d feet away from the Columbia Tower. Find the height of the Seafirst Tower. Give your answer to decimal places.

7 . If csc( θ ) =, epress tan( θ ) in terms of. π. Let f ( α) = cos( α) and g( α) = sin( α). Find the eact value of π f g.. Substitute = secθ into the epression π Assume 0 < θ <. 9 and simplify as much as possible. π. Suppose sinθ = A for < θ < π. What are the possible values of A? b) Find each in terms of A: π sin θ cotθ sin θ 5. On a day when the sun passes directly overhead at noon, a si foot tall man casts a shadow π of length Lt ( ) = 6 cot t where L is in feet and t is the number of hours since 6 a.m. Find eact values for the lengths of the shadow at 8:00 a.m., noon, and :00 p.m. b) Use your calculator to help you sketch an accurate graph of Lt () for 0 < t <. c) Determine the values of t at which the length of the shadow equals the man s height. 6. Find a possible formula for the functions graphed below. Use the general eponential form. b) Use a periodic function. y y 8 (-, 0.8) (, 8.75)

8 c) The vertical asymptote is = and the horizontal asymptote is y =. y 7. In each problem solve for the indicated variable on the given interval. Do not use your calculator. For t: cos ( t)sin( t) sin( t) 0 + = 0 t π b) For : cos = 0 sin 0 π c) For α : tan( α ) = 0 α π 9. In each case, determine if the epression is defined. If the epression is defined, simplify it as much as possible. arcsin b) cos ( π ) c) tan tan d) 5π arccos cos e) 0 tan arcsin 9 0. Let π + cos f( ) = 5 > +

9 Find f (0). b) Find such that f( ) = 0. c) Which of the following statements is true? (select all that apply) f( ) is continuous at =. f( ) has a jump (break) at =. f( ) has a hole at =. f( ) has a vertical asymptote at =.. Find the value of C that makes qt () continuous on (, ). Ct + t qt () = t + 8 t >. Determine if each function is continuous on the given interval. sinθ e π π g( θ ) = on, cosθ b) = on [, ] f( ). Use algebra to find the eact value of the following limits: 8z 7z t + 5t lim b) lim z z z 8 t 7 t + t. Use the graph of f( ) at the right to find the following: f (5) b) f () c) d) lim f( ) 5 lim f( ) f( )

10 5. Suppose f and g are functions such that lim f( ) = 8 and lim g ( ) =. Find the following: lim( f( ) 9) + b) g ( ) lim ( ) ( f ) c) lim cos( g ( )) 6. Find the following limits for f( ) =. + e lim f( ) b) lim f( ) c) lim f( ) + 0 d) lim f( ) 0 e) lim f( ) 0 7. Factor completely (simplify the factors): 5 0 z ( 6 z) + z ( 6 z) b) e + e 8. Simplify as much as possible: n b 5 n 5 b n+ m m 6 n+ b) y for m c) ln(5 e ) d) 5log 0 e) ( y+ ) yy ( + ) y + 9. Use algebra to solve for the indicated variable: For u: u + u = e b) For p: p + = 9 c) For : 5 9 = 0 7 d) For t: 5t 5t 7te t e = 0 e) For h : A= bh ( ) + h f) For w: log( w ) + log( w) = log(8) g) For : + y = h) For z: 5 + 5z z 0