Calculus Questions

Refer to the figure below to find the limit lim f x X 0 4 5 0 5 0 5 Q Select the correct choice below and fill in any answer boxes in your choice OA lim f x x 0 OB The limit does not exist

Refer to the graph at the right to find the limit lim 4 f x X 8 Select the correct choice below and fill in any answer boxes in your choice OA lim 4 f x 811 Simplify your answer B The limit does not exist 4 2

Use limits to compute f x f x 1 13x f x

Use limits to compute f x f x f x 9x X 2

Use limits to compute the derivative f 5 if f x 4x3 f 5 Simplify your answer

Suppose that lim f x 9 and lim g x 3 Compute the limits that exist X 4 X 4 a lim f x 3g x X 4 b lim f x g x X 4 c lim 4f x g x d lim X 4 f x x 4f x g x a Select the correct choice below and if necessary fill in the answer box to complete your choice OA lim f x 3g x X 4 Type an integer or a simplified fraction OB The limit lim f x 3g x does not exist X 4

Determine whether lim f x exists If so give the limit X 2 What is the value of the limit Select the correct choice below and if necessary fill in the answer box to complete your choice A The limit is OB The limit does not exist 3 2

36 T limsin 3x 0 1 0 01 0 001 0 0001 x 0 X 0 1 0 01 0 001 0 0001 sin 3x X a b C d X 0 1 0 01 0 001 0 0001 sin 3x X e f g h 37 Use the preceding two exercises to conjecture guess the value of the following limit lim positive real value sin ax x 0 x for a a

Let f x be a continuous function for which we know f 1 1 f 4 4 f 1 4 and f 4 16 Find the value of the definite integral gi f x z f x 1 dz a 2 2 2 1 dz Note as always do not round your answers

Let f x be a function with first order derivative f x second order derivative f x etc We know that So f x da 7 and f 0 1 f 1 2 f 0 3 f 1 4 f 0 5 f 1 6 Determine A 2 B 1 C 0 D 1 E 2 f x x dx

Ashton has an offer to buy an item with a sticker price of 4900 by paying 140 a month for 48 months Which of these groups of values plugged into the TVM Solver of a graphing calculator will give him the correct answer for the interest rate being offered A N 4 1 PV 4900 PMT 0 FV 140 P Y 1 C Y 12 PMT END B N 4 1 PV 0 PMT 4900 FV 140 P Y 1 C Y 12 PMT END C N 4 1 PV 0 PMT 4900 FV 6720 P Y 1 C Y 12 PMT END OD N 4 1 PV 4900 PMT 0 FV 6720 P Y 1 C Y 12 PMT END

Using the substitution usin 2 25 S where the constants K 25 x dx we obtain m K sin u cos udu m r 25 r dx C and n 0 Using this result and your knowledge about indefinite integrals of powers of sin u and cos u find the indefinite integ 2 Note Your answer should be in terms of x not u

Question 1 14 points Consider the function f x 2x 6 2x r3 a Factor the numerator and the denominator of f x as much as you can b Write the domain of f c Say whether f has any symmetries d Find the intersections of the graph of f with the z axis and the y axis if any e Find the subset of the domain on which f is positive f Compute the limits of f at any interesting point in R and at too Write the equation of any vertical horizontal or oblique asymptote of the graph of f g Compute the derivative f x h Find the stationary points of f if any Find the subsets on which f is increasing and decreasing Say whether the stationary points are relative minima maxima or neither Compute the values of f x at the stationary points Recall to factor a polynomial of the form ax bx c substitute t x so that it becomes at2 bt c and factor this first Then substitute back r for t and factor again i Draw the approximate graph of f encoding all the information obtained above

For the following exercises consider the function f x 1 x x 32 T Make a table showing the values of f for x 0 01 0 001 0 0001 0 00001 and for x 0 01 0 001 0 0001 0 00001 Round your solutions to five decimal places X 0 01 0 001 0 0001 0 00001 f x a b C d X 0 01 0 001 0 0001 0 00001 f x e f g h 33 What does the table of values in the preceding exercise indicate about the function f x 1 x x 34 To which mathematical constant does the limit in the preceding exercise appear to be getting closer

T In the following exercises set up a table of values and round to eight significant digits Based on the table of values make a guess about what the limit is Then use a calculator to graph the function and determine the limit Was the conjecture correct If not why does the method of tables fail 44 limsin 7 0 0 e 0 1 0 01 0 001 0 0001 sin 7 a b C d 8 0 1 0 01 0 001 0 0001 sin 7 e f g P

For the following exercises consider the function f x x 1 x 1 30 T Complete the following table for the function Round your solutions to four decimal places X 0 9 0 99 0 999 0 9999 f x response a b C P d X 1 1 1 01 1 001 1 0001 f x e f g h 31 What do your results in the preceding exercise indicate about the two sided limit limf x Explain your x 1

A Lorenz curve is given by y L x where 0 x 1 represents the lowest fraction of the population of a society in terms of wealth and 0 sys1 represents the fraction of the total wealth that is owned by that fraction of the society Use this information to complete parts a through g Click the icon to view an example of a Lorenz curve Now show that 2A 1 2 L x dx First rewrite the expression 1 2 L x dx in terms of B 1 1 2 L x dx 1 2B Substitute the expression for B in terms of A found in a previous step for B and then simplify 1 2 0 D 1 2 24 1 1 A Therefore G 2A 1 2 L x dx by the O Substitute Simplify transitive property completing the proof e Compute the Gini index for the cases L x x and p 1 3 1 9 2 3 4 The Gini index for p 1 3 is 0 130 Type an integer or decimal rounded to three decimal places as needed The Gini index for p 19 Type an integer or decimal rounded

the following exercises set up a table of values to find the indicated limit Round to eight digits x 4 x 2x x 6 38 lim X 1 9 1 99 1 999 1 9999 x 4 x x 6 a b C d X 2 1 2 01 2 001 2 0001 x 4 x x 6 e f g h

ercises 65 74 Use the graph of f to determine intervals where fis creasing and where f is decreasing 65 3 1 T 321 2 23 2 3 23 X

What is the net signed area between f x 2x 5 and the x axis over the interval 1 6 Do not include any units in your answer

a A radioactive material decays according to the formula w 7 64 e 0 05 t 1 Find the half life of the material 2 How long will it take until the substance to decay to 25 of its initial amount

Find the area of the region described The region in the first quadrant bounded by y 5 and y 5 sin x on the interval The area of the region is Type an exact answer using x as needed T

A Lorenz curve is given by y L x where 0 x 1 represents the lowest fraction of the population of a society in terms of wealth and 0 y 1 represents the fraction of the total wealth that is owned that fraction of the society Use this information to complete parts a through g Click the icon to view an example of a Lorenz curve a A Lorenz curve y L x is accompanied by the line y x called the line of perfect equality Explain why this line is given this name Choose the correct answer below OA The lowest 50 of households own 60 of the wealth OB The lowest 50 of households own 100 of the wealth OC The lowest p of households owns exactly 1 p of the wealth for 0 p 100 D The lowest p of households owns exactly p of the wealth for 0 sp 100

Perform a first derivative test on the function f x x 5x 2 5 5 a Locate the critical points of the given function b Use the first derivative test to locate the local maximum and minimum values c Identify the absolute maximum and minimum values of the function on the given interval when they exist Simplify your answer OB There is no local minimum 2 Locate the maximum value Select the correct choice below and if necessary fill in the answer box within your choice OA There is a local maximum at x Simplify your answer B There is no local maximum A c Identify the absolute minimum value of the function Select the correct choice below and if necessary fill in the answer boxes to complete your ch The absolute minimum value is Simplify your answer OB There is no absolute minimum 33 4 Identify the absolute maximum value of the function Select the correct choice below and if necessary fill in the answer box within your choice OA The absolute maximum value is Simplify your answer OB There is no absolute maximum

Find the area of the shaded region shown in the figure which is not shown to scale COL

Perform a first derivative test on the function f x x 5x 2 5 5 a Locate the critical points of the given function b Use the first derivative test to locate the local maximum and minimum values c Identify the absolute maximum and minimum values of the function on the given interval when they exist Simplify your answer Use a comma to separate answers as needed OB The function does not have a critical point b Locate the minimum value Select the correct choice below and if necessary fill in the answer box within your choice There is a local minimum at x TE 5 2 Simplify your answer OB There is no local minimum Locate the maximum value Select the correct choice below and if necessary fill in the answer box within your choice OA The absolute minimum value is Simplify your answer OA There is a local maximum at x Simplify your answer B There is no local maximum c Identify the absolute minimum value of the function Select the correct choice below and if necessary fill in the answer boxes to complete your

Find the area of the following region expressing your result in terms of the positive integer n 2 The region bounded by f x x and g x x n for x 0 The area of the region in terms of n is

Find the area of the shaded region shown in the graph CE 20 y 20 sin x y 20 cos x R 2 O

Find the net signed area of f x X the interval 0 6 2 over

Given the function f x 3x 10 find the net signed area between f x and the x axis over the interval 6 2 Do not include any units in your answer

Directions Use the function f x below to answer the following questions Part 1 Intervals Find the open interval s on which f is increasing decreasing 1 Increasing 2 Decreasing f x 7x 4 Answer Formats If necessary use INF and INF to denote and o If the function is not increasing decreasing for any interval enter NONE If there is more than one interval type your answer as a comma separated list For example a b c d Part 2 Relative Extremum Find the x values s where f has a relative maximum relative minimum

Directions The following function f x has one relative maximum and one relative minimum point Find these points using the First Derivative Test and a sign chart f x x 27x Part 1 Critical Numbers 1 Find the derivative f x 2 List the z values where f x 0 separate by commas Part 2 Intervals of Increase and Decrease Since there are two critical numbers there should be three intervals to consider on our sign chart State the indicated interval below and whether the function is increasing or decreasing there Help with entering help intervals 1 The left most interval is and there the function is inc or dec 2 The middle interval is 3 The right most interval is and there the function is inc or dec and there the function is inc or dec

Part 1 Intervals of Concavity 1 Find the open interval s on which f is concave up 2 Find the open interval s on which f is concave down f x x 8x 24x x 6 Part 2 Inflection Points 1 Find the x values s where f has an inflection point Answer 0 00 Answer Formats If necessary use INF and INF to denote and o If the function does not have any enter NONE If the function has more than one type your answer as a comma separated list For example a1 b a2 b Answer none Answer

Directions Consider the following function f a at a 2 Part 1 Difference Quotient If h0 then the difference quotient can be simplified into the form 2 A f 2 h where A B and C are all non negative constants Find these constants A 2 and B49 and C 14 Note It s possible for one or more of these constants to be 0 1 f x 2x 3 2 f 2 49 Part 2 Derivative Use the simplified expression from Part 1 to then calculate the derivative f 2 B Ch f 2 49 B Ch f 2 lim A A 2 and B 49 and C 14 Note It s possible for one or more of these constants to be 0 That is note the negative in front f 2 h f 2 h Part 2 Derivative Use the simplified expression from Part 1 to then calculate the derivative f 2 f 2 lim h0 Part 3 Tangent Line Find the equation of the tangent line to the curve y f a at a 2 Tangent Line y f 2 h f 2 h

Directions Use the equation given below to answer the following question Derivative Use implicit differentiation to calculate da dy Note that we are calculating the derivative of z with respect to y the opposite of what we normally dy This means we need to regard y as the independent variable and a as the dependent variable x y xy 2xy 0 da

Directions Use the equ Part 1 Derivative dy Calculate the derivative by implicit differentiation da dy da Part 2 Evaluation Evaluate the derivative at the point 0 2 dy da 0 2 x 4xy y 4

Find an equation of the tangent line to the given function at a 2 y x 4 6 f x x X

7 Determine the values of the trigonometric functions of a where sin a 25 tan Type a simplified fraction and cos a 24 25

Given the function f x 4x 8 find the net signed area between f x and the x axis over the interval 5 6 Do not include any units in your answer Provide your answer holow

Use the figure and any appropriate relationship s among circular functions to find the value tan 5x 6 Select the correct choice below and if necessary fill in the answer box to complete your choice O A 5x 6 Simplify your answer including any radicals Use integers or fractions for any numbers in the expression OB The expression is undefined tan 12 12 1 13 22 12 12 7 3x 5x4 6 1 0 7x 13 2x 65x 4 w 4x 3 Ay 0 1 5x 3 4 I 60 11x 1 0 7x6 4 0 1 3x 2 x y 1 N

Use the given figure and any appropriate relationship s among circular functions to find 3 sin 4 0 1 14 4 4 1 0 Ay 4 4 11x 0 1 7x 6 5x 4 1 0 43 2 oo 4x sin 3 Type an integer or a fraction Type an exact answer using radicals as neede

Use the following figure and any appropriate relationship s among circular functions to 9x find tan 4 4 1 0 10 4 1 1 43 WIM 4 G 0 1 1 0 X 4 1 5 4 4 9x tan 4 Type an exact answer using radicals as needed

Use the x y coordinates in the figure to find the value of the trigonometric function at the indicated real number s or state that the expression is undefined COS 53 4 1 3x 5x 4 6 1 0 7x 6 5x 1 4 1 12 47 on 14 0 1 4x 3 13 7 2 Ay IN 3 5x 3 3x 2 0 1 x 1 AI x 4 7x 6 12 N 6 0 11x 1 0 13 2 4 Select the correct choice below and if necessary fill in the answer box to complete your choice O A 5x 3 Simplify your answer including any radicals Use integers or fractions for any numbers in the expression COS OB The expression is undefined

Use the x y coordinates in the figure to find the value of the trigonometric function at the indicated real number s or state that the expression is undefined 7x 6 sin 12 12 1 7 on 7 0 1 2 2 3x 5x 4 6 1 0 7x 4 1 65x 4 4 4 2x 4x 3 k K 2 3x KIM K 4 5x 3 1 2 0 1 7x6 4 6 0 11x 12 12 712 2 1 0 13 12 4 4 4 OU HULING Select the correct choice below and if necessary fill in the answer bc complete your choice OA 7x sin 6 Simplify your answer including any radicals Use integers or for any numbers in the expression OB The expression is undefined

Use a calculator in radian mode to find approximations for coss and sins for the number s 3 86 These are NOT exact values they are only approximations Then determin the quadrant in which the point on the unit circle corresponding to s lies Finally find approximations for tan s cots secs and csc s sin 3 86 Round to three decimal places ACCED

Use a calculator to approximate sin 5 8916 and cos 5 8916 sin 5 8916 Round to four decimal places

estion What is the net signed area between f x 2x 3 and the x axis over the interval 4 2 Do not include any units in your answer Provide your answer below 11262

Question Given the function f x x 2 find the total area between f x and the x axis over the interval 5 1 Provide your answer below

Question Find the total area between the function f x 2x and the x axis over the interval 3 3 Provide your answer below

Determine the point on the graph of the unit circle that I I 2 corresponds to Then findcos andsin and state 2 which functions are undefined for I cos 2 KIN 0 2 I 2 The point on the unit circle is 0 1 Type an ordered pair Type an integer or a simplified fractio T T Simplify your answer sin 1 Simpl 2 1 0 2 cotangent tangent Ay 0 1 0 0 1 1 0 O