Student study guide for the MAT 151 Spring 2016 final examination

Transcription

1 Student study guide for the MAT 151 Spring 016 final examination Use the problems in this study guide to help you prepare for the problems on the final. The problems below are similar to the ones on the final, and include other problems types which you should know how to solve (but which are not on the test.) The actual final examination has approximately twenty-five problems. ** Solve. Use interval notation to express your answer. Then draw and label a real number line to graph the solution. 4 5x 6 ** Write a slope-intercept equation for the line which passes through the points ( 1, ) and (1,4). ** i) Write the equation of the horizontal line which passes through (6, 5). ii) Now write the equation of the vertical line which passes through (6, 5). ** Give the domain and range of the function State your answers using interval notation. Domain = 5 x 1 T( x). 4 x Range =

2 ** Find the equation of the line which passes through (3,7) and is perpendicular to the line x3y 4. ** Suppose that $300 is invested into an account that pays interest quarterly at the rate of 1.75% per year. i) How much money will be in the account after 9 years? ii) How much money will be in the account after 9 years if interest is compounded continuously? iii) If interest is compounded continuously, what will be the doubling time? (In other words, how long will it take the account to grow to $4600?) ** Data on sales, y, in the United States for pets, pet products, and related services are given in the table below for various years, x. Year, x Expenditures, y (billions of dollars) 1991, , , , , , , i) Use a graphing calculator to find the line of best fit (i.e., the linear regression equation). ii) Find the correlation coefficient (i.e., the residual) r. Then decide whether the line is a good fit to the data and provide a reason for your decision. r Good fit? iii) What would be the expected expenditure amount for the year 015? (Round your answer to the nearest tenth.)

3 ** The number of used jets for sale worldwide was 10 in In 009 that number was [source: UBS Investment Research] Find the average rate of change in the number of used jets for sale over this period (from 1999 to 009). ** Use the functions f(x) and g(x) to find the indicated functions and function value. x f ( x), g( x) 3x x i) ( fg)( x) ii) ( f g)( x) iii) ( g f)( 3) ** Find the following indicated function values x x 5 W ( x) 3 x 5 x 4 x x 4 W(6) W( 7) W( 1) ** Choose whether the given function g(x) is EVEN, ODD, or NEITHER. (Circle one choice.) 1 gx ( ) x Now explain or justify your answer.

4 ** The power P expended by heat in an electric circuit of fixed resistance varies directly as the square of the current C in the circuit. A circuit expends 180 watts when a current of 6 amperes is flowing. What is the amount of heat expended when the current is 10 amperes? ** i) Write an equation for a function f(x) which transforms J( x) 5 3x by stretching J(x) horizontally by a factor of 3. f( x) ii) Write an equation for a function g(x) which transforms J( x) 5 3x by translating (shifting) J(x) 6 units to the left. gx ( ) iii) Write an equation for a function h(x) which transforms J( x) 5 3x by reflecting J(x) across the y-axis. hx ( ) ** i) Find the value of the discriminant for the quadratic equation 3y 5y 9. ii) Now, based on this number, describe the types of solutions this equation has: ** Solve the quadratic equation and clearly show all your steps. 3y 5y 9

5 ** The profit function P(x) for a business is found by using the formula P( x) R( x) C( x), where R(x) and C(x) are the revenue and cost functions, respectively. The Arthur s Bakery has revenue and cost functions given by R( x) 36x 0.4 x, C( x) 13 6x Find the maximum profit. Then find the number of units that must be sold to yield this maximum profit. ** An open box is made from a 15.5-in by 13.5-in piece of cardboard. A square is cut from each corner and then folding up the edges. If the area of the resulting base is 143 square inches, find the length of the sides of the squares. ** Solve. Show your work. 5 4 t 3 t t 4 t ** Solve. Show your work. x1 x 3 ** Express your answer using interval notation. 3x 5 8

6 ** Divide and simplify. Express your answer in standard form (a + bi) ** Solve the polynomial inequality. Express your answer using interval notation. x 3 x ** Solve the rational inequality. Express your answer using interval notation. x1 x3 x5 x ** Provide the following information about the polynomial function. m( x) 7 3x 4 1.6x 3 Leading term Leading coefficient End behavior Classify the function type. Circle one of the choices below. Radical Rational Constant Cubic Exponential Quartic Quadratic Linear ** i) Convert the equation to an equivalent exponential equation. log 3 15 t x ii) Convert the equation to an equivalent logarithmic equation. 5z b 4

7 ** Provide the indicated information about the rational function rx ( ) 1 x x 8x6 i) Domain (using interval notation) ii) Equations of all asymptotes iii) x-intercepts iv) y-intercepts v) Does the graph contain a hole (missing single point)? ** Using a graphing calculator find the indicated values for the polynomial function. Where appropriate, express values to three places beyond the decimal (e.g., 1.64). P x x x x 4 3 ( ) 3 5 Real zeros Relative maxima Range Relative minima ** i) Make an accurate sketch of the graph of the function ii) Is this function one-to-one? x U( x) 3 x iii) If so, show the steps to find a formula for the inverse function. ** The following values of the logarithms apply for a certain base b. logb3 0.61, logb , logb

8 Use these values to find the following values of the logarithms (for that same base). log b logb 0. logb 15 logb 7 ** Solve. Show your work. 1 log( x1) log( x 1) ** Solve the exponential equation. First provide an exact answer (containing logarithms). Then express that answer as a decimal approximation (rounded to two places beyond the decimal). x1 x ** The table provides data on the number of major hurricanes in recent years. Year # of hurricanes

9 i) Use your calculator to find the linear regression equation ( y ax b ) which provides the best fit to the data. Round constants to three places beyond the decimal. ii) Use your calculator to find the quadratic regression equation ( y ax bx c ) which provides the best fit to the data. Round constants to three places beyond the decimal. 3 iii) Use your calculator to find the cubic regression equation ( y ax bx cx d ) which provides the best fit to the data. Round constants to three places beyond the decimal. iv) Now use the equation with the highest Round your result to the nearest integer. R value to predict the number of major hurricanes in 00. ** A dietician must plan a breakfast menu that provides 460 Calories, 9 grams of fat, and 55 milligrams of calcium. One plain bagel contains 00 C, g of fat, and 9 mg of calcium. A onetablespoon serving of cottage cheese contains 100 Cal, 10 g of fat, and 4 mg of calcium. One banana contains 105 Cal, 1 g of fat, and 7 mg of calcium. How many servings of each food type are required to provide the desired nutritional values? ** A chilled gelatin salad that has a temperature of 43 F is taken from the refrigerator and placed on a table in a room with an air temperature of 68 F. After 1 minutes the temperature of the salad is 55 F. What will the temperature of the salad be 0 minutes after being removed from the refrigerator? Hint: the formula expressing Newton s law of cooling may be helpful. () T t T T T e kt 0 1 0

10 ** The population of Papua New Guinea was million in 011 and was growing with an exponential growth rate of 1.985% per year. i) Write an exponential function P(t) for the population function. ii) Now find the population for 015 (rounded to the nearest tenth of a million). iii) Find the doubling time. iv) Find the year in which the population will reach 10 million. ** Find the solution, if any, to the following linear system. x 3y z 11 x 3y 4z 4 3x y 3z 4 ** Consider the three matrices A, B, and C A 6 1 3, B 7 3 1, C For the six indicated matrices below, find all which are defined. i) BC= ii) BC = iii) C+B= iv) B-4A = v) C -1 = vi) B -1 =

11 ** Fill in the four blanks to satisfy the following matrix equation ** Express the following sum by writing it using sigma notation ** Find the value for the following sum ** Predict the general (i.e., the n th ) term by writing a formula for an. 7 / 8,7 /, 14,56, 4,... ** i) For the following arithmetic series, find the twelfth term, a1. 17,1,7,, 3,... ii) For the following geometric series, find the twelfth term, a , 0.5, 0.5, 1,...

12 ** Find the sum of the infinite series, if it exists. 0, 40 / 3,80 / 9, 160 / 7... ** Find the sum of the infinite series, if it exists. 6, 9, 13.5, 0.5,... ** For a geometric series 1 r and S5 31. Find the values of the following two terms in the series. i) a1 ii) a5