Application of derivatives Questions and Answers

Express as a sum or difference of logarithms without exponents 5 log c 9 X 5 9 y z The answer is Simplify your answer Use integers or fractions for any numbers in the expression

Match the function with its graph I E III a f x x b f x x c f x x ya ya Select V Select Select V II IV ya

1 01 1 2 dx lim b O True O False dx 1 Click here to view the transcript

Explain why the integral is improper 36 1 0 x dx X 1 6 1 4 1 2 0 8 0 6 0 4 0 2 5 10 15 20 25 30 35 O At least one of the limits of integration is not finite O The integrand is not continuous on 0 36 Determine whether it diverges or converges O converges O diverges Evaluate the integral if it converges If the quantity diverges enter DIVERGE

se graphical methods to solve the following linear programming problem Minimize z 3x 2y subject to x ys 10 3x 2y26 x20 y20 Graph the feasible region using the graphing tool to the right The minimum value of z is at the corner point Simplify your answers KIB 16 14 12 10 8 6 2 8 FN 6 00 8 10 12 LYZ

10 10 SEOR YEAR 17 5 55 1111 1119 Enable Zoom Pan y 10 30 n 240 10 X 31 9 10 10 and x 10 79 1 5 ES 921 Enable Zoom Pan ETIRE Zoom Pan 100

Question Choose the graph of the inverse function Hint The line y x is drawn in blue in the answer choices 6 N N A 4 For more information on this topic look in section 1 8 of your textbook especially at Examples 5 6 and 7 You might also want to look at the Graph Transformations presentation 2 N O N 4 60 6

Solve the system of two linear inequalities graphically Step 3 of 3 Select and graph the option that describes the overall solution set Answer Select the region you wish to be shaded x 1 or y 5 the union of the individual solution sets the intersection of the individual solution sets 10 45 2012 A Enable Zoom Pan

Use this definition to find f a at the given number a f x 3x 1 a 8

Consider the following absolute value inequality Step 1 of 2 Rewrite the given inequality as two linear inequalities Answer 17y 712 45 Enter the two inequalities in the boxes and select the proper word to join the inequalities Inequalities and Oor

Insert C or in the blank to make the statement true 21 b d j b d j f p

VA Sketch the graph of f on the same coordinate axes y e y VA

ind the corner points of the feasible region of the linear programming problem 10 Maximize z 3x 4y subject to x y 6 2x y 8 x 0 y 0 The feasible region is given below 6 4 2 S S 20 10 X

Consider the following curve y 3x 7x 1 Find the slope m of the tangent line at the point 3 7 m Find an equation of the tangent line to the curve at the point 3 7 y

7 6 5 4 3 24 6 5 4 4321 3 2 1 1234567 1 Select the correct answer below O x 4 with multiplicity 2 and x 1 with multiplicity 3 x 4 with multiplicity 1 and x 1 with multiplicity 2 x 4 with multiplicity 3 and x 1 with multiplicity 2 x 4 with multiplicity 1 and x 1 with multiplicity 4 x 4 with multiplicity 4 and x 1 with multiplicity I 1 2 3 4 5 6 7

Based on the graph of f x shown below what is the degree of f x Drovide our answer helow 7654321 5 4 LLL 7 6 5 4 3 2 11 1234567 4567DENT 2 3 4 5 6 H 7

the problem 8 Burger Barn s three locations sell burgers fries and soft drinks Barn I sells 900 burgers 600 order of fries and 750 soft drinks each day Barn II sells 1500 burgers a day and Barn III sells 1150 Soft drink sales number 900 a day at Barn II and 825 a day at Barn III Barn II sells 950 orders of fries per day and Barn III sells 800 The daily sales for all locations are shown in the following matrix Barn I Burgers 900 Fries 600 Drinks 750 Burgers Burgers cost 1 50 each fries 0 90 an order and soft drinks 0 60 each The costs at all locations are shown in the following matrix 1 50 Barn II 1500 950 900 Fries 0 90 Barn III 1150 800 825 Drinks 0 60

Find the value of the objective function at each corner point of the feasible region and determine the maximum and the minimum 11 Objective function z 4x 3y Corner Points of a bounded feasible region 0 0 0 2 and 66 5 5

Which of the following are monomials Select all correct answers Select all that apply 2a b 3a bc 2a 2a b 3a bc 3abc 2a 2ab c

simplex table a list the basic and the nonbasic variables b find the basic feasible solution determined by setting the nonbasic variables equal to 0 and c decide whether this is a maximum solution a What are the basic variables Use a comma to separate answers as needed What are the nonbasic variables x S X1 X2 X3 S S2 3 0 1 0 1 0 00 Use a comma to separate answers as needed b Find the basic feasible solution by setting the nonbasic variables equal to 0 1 X 1 1 X3 Z c Is this solution a maximum O Yes O No 2 2 4 4 3 1 3 S1 S 11 N 0 28 0 15 1 12

Find the leading coefficient of the following polynomial 7 2x 3x x

person age This situation describes a function y f x where the two variables are the person s age in years and the person s height in inches Which variable is age and which variable is height What would you expect the range of the function to be if Dom f 0 85 Ran f

t up the system of equations Do not solve 1 An investment firm recommends that a client invest in bonds rated AAA A and B The average yield on AAA bonds is 6 on A bonds 7 and on B bonds 8 The client wants to invest twice as much in AAA bonds as in B bonds How much should be invested in each type of bond if the total investment is 25 000 and the investor wants an annual return of 1800 on the three investments

A function s inverse when it exists switches the roles c x and y If a function is a machine then the function s inverse is the same machine run in reverse

Perform the indicated operations where possible 22 3 2 1 2 6 5 3 4 6 3 54 4

What would the x intercept of the function tell you in terms of age and height Would you expect such a poin to exist The x intercept tells us I expect such a point to exist because

Write a system of inequalities that describes all the given conditions and graph the feasible region of the system Joyce is the marketing director for a new company selling a fashion collection for young women and she wishes to place ads in two magazines magazine V and magazine E Joyce estimates that each one page ad in magazine V will be read by 1 8 million people and each one page ad in magazine E will be read by 1 3 million people Joyce wants to reach at least 8 million readers and to place at least 2 ads in each magazine Write a system of inequalities that describes all the given conditions Let x represent the number of ads in magazine V and y the number of ads in magazine E Choose the correct answer below

2 in a straight line is given by s a Find the average velocity in ft s over each time interval 1 4 8 3 ii 6 8 2 iii 8 10 0 iv 8 12 1 ft s ft s S ft s ft s b Find the instantaneous velocity in ft s when t 8 X ft s 9t 18 where t is measured in seconds c Draw the graph of s as a function of t and draw the secant lines whose slopes are the average velocities in part a Then select the graph of s with the tangent line whose slope is the instantaneous velocity in part b S

a Find the slope m of the tangent to the curve y 2 x at the point where x a m b Find equations of the tangent lipes at the following points 25 10 y y 16 8 c Graph the curve and both tangents on a common screen y 5 10 O 15 10 20 30 40 X y 15 10 5 10 20 30 40 X y 15 10 10 20

Recall the constant multiple law which states that if c is a constant and the limit lim f x exists then the following holds 818 1 cf x c c lim f x x lim X18 Applying this limit law gives us the following 72 8 m lim 1 5 lim lim 5 lim G X18 x48 X18 8 lim x18 3 lim x18

z Find the value of at the point 1 1 1 if the equation 8xy z x 4yz 5 defines z as a function of the two ax independent variables x and y and the partial derivative exists z ax 1 1 1 Type an integer or a simplified fraction

Find an equation for the level curve of the function f x y that passes through the given poin f x y 49 x y 2 3 3 3 An equation for the level curve is Type an equation

Rewrite sin x 5 a Enclose arguments of functions in parentheses For example sin 2x 57 6 sin x 57 in terms of sin x and cos ac

Sketch the graph of the function Identify any local extrema and points of inflection x 25 f x O A What are the coordinates of the relative maxima Select the correct choice below and if necessary fill in the answer box to complete your choice X O A Simplify your answer Type an ordered pair Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed OB There is no maximum What are the coordinates of the relative minima Select the correct choice below and if necessary fill in the answer box to complete your choice COO A Simplify your answer Type an ordered pair Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed OB There is no minimum What are the coordinates of the points of inflection Select the correct choice below and if necessary fill in the answer box to complete your choice OA Simplify your answer Type an ordered pair Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed B There are no inflection points Choose the correct graph below B O C D

Find the locations of the absolute extrema of the function on the given interval 2 f x 3 2 20x 13 0 5 The absolute minimum occurs at x The absolute maximum occurs at x Type exact answers

Find the location of the absolute minimum extrema for the function 2 2 A 2 0 OB 3 2 OC 2 0 OD None F 2 d

An open top rectangular box is being constructed to hold a volume of 300 in The base the box is made from a material costing 6 cents in The front of the box must be decorated and will cost 11 cents in The remainder of the sides will cost 3 cents in Find the dimensions that will minimize the cost of constructing this box Front width Depth Height in in in

f x x 8x 7 5 The Rolle s Theorem applies to the function closed interval 7 Find all values of c in the open interval 1 7 such that f c 0 on the

7 For the function f x 4x 30x 2 a Find the critical numbers of f if any b Find the open intervals where the function is increasing c decreasing and c Identify all relative extrema

3 Locate the absolute extrema of the given function on the closed interval 10 10 f x 10x x 25

10 Find all relative extrema of the function which are maximum if any and which are minimum if any f x 3x 24x 52 Indicate

15 A rectangular page is to contain 144 square inches of print The margins on each side are 1 inch Find the dimensions of the page such that the least amount of paper is used

6 The Mean Value Theorem can be applied to the function f x x on the closed interval 3 11 Find all numbers c in the open interval 3 11 f c 11 f 3 11 3 such that

9 Find all points of inflection on the graph of the function f x x 6x

8 Determine the open intervals on which the graph of f x 5x 3x 1 is concave downward or concave upward

Sketch the graph of the given function by determining the appropriate information and points from the first and second derivatives y 4x 48x 4 vvnat are the coordinates or the relative maxima Select the correct choice below and if necessary fill in the answer box to complete your choice A 2 60 Simplify your answer Type an ordered pair Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed B There is no maximum What are the coordinates of the relative minima Select the correct choice below and if necessary fill in the answer box to complete your choice A 2 68 Simplify your answer Type an ordered pair Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed OB There is no minimum What are the coordinates of the points of inflection Select the correct choice below and if necessary fill in the answer box to complete your choice OA Splify your answer Type an ordered pair Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed OB There are no inflection points

A casino hotel manager has approximated demand for two room suites to be p 9000 2 where p is the price in dollars and g is the quantity demanded Use implicit differentiation to find and dq interpret when q 5 dp Choose the correct answer below and fill in the answer box to complete your answer OA da O ge ge dp B dq q 20 dp which means that if the price is increased by 1 the quantity demanded decreases by approximately which means that if the price is increased by 1 the quantity demanded increases by approximately gs gs dq dp dq dp suites suites

Find the equation of the tangent line at the given point on the following curve x y 5 1 2 The equation of the tangent line to the point 1 2 is y

Find for the equation x y 2 using implicit differentiation dx

For a certain automobile M x 015x 1 21x 7 3 30 x 60 represents the miles per gallon obtained at a speed of x miles per hou a Find the absolute maximum miles per gallon and the speed at which it occurs b Find the absolute minimum miles per gallon and the speed at which it occurs This occurs at mph a The absolute maximum miles per gallon is Type an integer or decimal rounded to the nearest thousandth as needed