Application of derivatives Questions and Answers
![Each expression simplifies to a constant a single trigonometric function or a power of a single trigonometric function Use fundamental identities to simplify each expression 1 sin x tan x cos x 2 sec x cos x 3 tan x cos x 4 sec x 1 5 tan x 2 sin x csc x CA 9 G](https://media.kunduz.com/media/sug-question-candidate/20231206221425910944-4704102.jpg?w=256)
Calculus
Application of derivativesEach expression simplifies to a constant a single trigonometric function or a power of a single trigonometric function Use fundamental identities to simplify each expression 1 sin x tan x cos x 2 sec x cos x 3 tan x cos x 4 sec x 1 5 tan x 2 sin x csc x CA 9 G
![Evaluate the following expressions Your answer must be in radians written as multiple of 7 Note that is already provided in the answer so you just have to write the appropriate multiple E g if 2 is the answer you should write 1 2 Do not use decimals Your answer should be a fraction or an integer a tan 1 b tan 3 c tan 3](https://media.kunduz.com/media/sug-question-candidate/20231206213532715428-4704102.jpg?w=256)
Calculus
Application of derivativesEvaluate the following expressions Your answer must be in radians written as multiple of 7 Note that is already provided in the answer so you just have to write the appropriate multiple E g if 2 is the answer you should write 1 2 Do not use decimals Your answer should be a fraction or an integer a tan 1 b tan 3 c tan 3
![Evaluate the following expressions Your answer must be an angle T 2 0 in radians Give exact answers do not use decimal answers You can enter the number 7 as pi E g if the answer is 2 you could enter pi 2 or 1 2 pi sin 1 sin cos 1 os 1](https://media.kunduz.com/media/sug-question-candidate/20231206213511375295-4704102.jpg?w=256)
Calculus
Application of derivativesEvaluate the following expressions Your answer must be an angle T 2 0 in radians Give exact answers do not use decimal answers You can enter the number 7 as pi E g if the answer is 2 you could enter pi 2 or 1 2 pi sin 1 sin cos 1 os 1
![oblem 3 Evaluate the following integral TT 3 6 cot x In sin x dx In 4 In 3 2 2 Problem 3 In 4 In 3 2 Just Save Problem 3 Enter your ansv as in these exa Submit Problem 3 for Grading Attempt 1 Attempt 2 At](https://media.kunduz.com/media/sug-question-candidate/20231206212331783709-4270279.jpg?w=256)
Calculus
Application of derivativesoblem 3 Evaluate the following integral TT 3 6 cot x In sin x dx In 4 In 3 2 2 Problem 3 In 4 In 3 2 Just Save Problem 3 Enter your ansv as in these exa Submit Problem 3 for Grading Attempt 1 Attempt 2 At
![Fill in the blank to complete the trigonometric identity Simplify your answer complete COS](https://media.kunduz.com/media/sug-question-candidate/20231206201427476075-5836419.jpg?w=256)
Calculus
Application of derivativesFill in the blank to complete the trigonometric identity Simplify your answer complete COS
![Use an identity to find the exact value of each expression Note You are not allowed to use decimals in your answer sin 2 T 3 3 COS os 7 6 tan TT TT F BI A](https://media.kunduz.com/media/sug-question-candidate/20231206200227011671-4704102.jpg?w=256)
Calculus
Application of derivativesUse an identity to find the exact value of each expression Note You are not allowed to use decimals in your answer sin 2 T 3 3 COS os 7 6 tan TT TT F BI A
![Find the sum 5 i 0 41](https://media.kunduz.com/media/sug-question-candidate/20231206194409050257-5836419.jpg?w=256)
![In the triangle below x 8 Use the Law of Cosines to solve the triangle Round your answers to two decimal places B 12 cm O O O B 18 cm x cm C](https://media.kunduz.com/media/sug-question-candidate/20231206194251424497-5836419.jpg?w=256)
Calculus
Application of derivativesIn the triangle below x 8 Use the Law of Cosines to solve the triangle Round your answers to two decimal places B 12 cm O O O B 18 cm x cm C
![A stone was dropped off a cliff and hit the ground with a speed of 80 ft s What is the height of the cliff Use 32 ft s for the acceleration due to gravity ft](https://media.kunduz.com/media/sug-question-candidate/20231206174647768876-5061986.jpg?w=256)
Calculus
Application of derivativesA stone was dropped off a cliff and hit the ground with a speed of 80 ft s What is the height of the cliff Use 32 ft s for the acceleration due to gravity ft
![5 7C5](https://media.kunduz.com/media/sug-question-candidate/20231206165201425774-5836419.jpg?w=256)
![1 2 3 6 700 6 2 W 4 70 3 1 5 g x 2 1700 J 6 3 N 0 2 The functions and g have continuous second derivatives The table above gives values of the function their derivatives at selected values of x a Let k x f g x Write an equation for the line tangent to the graph of k at x 3](https://media.kunduz.com/media/sug-question-candidate/20231206162448915606-6339451.jpg?w=256)
Calculus
Application of derivatives1 2 3 6 700 6 2 W 4 70 3 1 5 g x 2 1700 J 6 3 N 0 2 The functions and g have continuous second derivatives The table above gives values of the function their derivatives at selected values of x a Let k x f g x Write an equation for the line tangent to the graph of k at x 3
![2 2 20 Points Find the following antiderivatives ln x](https://media.kunduz.com/media/sug-question-candidate/20231206155126190407-4971328.jpg?w=256)
![Find the value of each trigonometric ratio 33 sin A A Z 36 35 cos Z B 15 C 39 28 35 Y 21 X 34 sin Z Z 24 25 Y 7 X Z 36 cos X 36 45 1 25 Y 27 X](https://media.kunduz.com/media/sug-question-candidate/20231206045942494729-4582308.jpg?w=256)
Calculus
Application of derivativesFind the value of each trigonometric ratio 33 sin A A Z 36 35 cos Z B 15 C 39 28 35 Y 21 X 34 sin Z Z 24 25 Y 7 X Z 36 cos X 36 45 1 25 Y 27 X
![Find the value of each trigonometric ratio 62 cos C 32 C B 64 tan Z Z 27 Y 40 24 45 36 A X 63 tan Z X 18 65 sin C C 14 B Y 30 24 50 48 Z A](https://media.kunduz.com/media/sug-question-candidate/20231206045555769880-4582308.jpg?w=256)
Calculus
Application of derivativesFind the value of each trigonometric ratio 62 cos C 32 C B 64 tan Z Z 27 Y 40 24 45 36 A X 63 tan Z X 18 65 sin C C 14 B Y 30 24 50 48 Z A
![Convert each radian measure into degrees 5 18 1 Convert each degree measure into radians 3 45 2 10T 3 4 300](https://media.kunduz.com/media/sug-question-candidate/20231206045525358498-4582308.jpg?w=256)
Calculus
Application of derivativesConvert each radian measure into degrees 5 18 1 Convert each degree measure into radians 3 45 2 10T 3 4 300
![A cell receives nutrients through its surface and its surface area is proportional to the two thirds power of its weight Therefore if w t is the cell s weight at time t then w t satisfies w aw2 3 where a is a positive constant t Solve this differential equation with the initial condition w 0 7 initial weight 7 units w t 3 at 7 X](https://media.kunduz.com/media/sug-question-candidate/20231206033730227043-5824775.jpg?w=256)
Calculus
Application of derivativesA cell receives nutrients through its surface and its surface area is proportional to the two thirds power of its weight Therefore if w t is the cell s weight at time t then w t satisfies w aw2 3 where a is a positive constant t Solve this differential equation with the initial condition w 0 7 initial weight 7 units w t 3 at 7 X
![Use the Laws of Logarithms to combine the expression 7 log x 8 log x 1](https://media.kunduz.com/media/sug-question-candidate/20231206032016568264-4729556.jpg?w=256)
Calculus
Application of derivativesUse the Laws of Logarithms to combine the expression 7 log x 8 log x 1
![Find the function of the form y log x whose graph is given y II 1 0 0 5 0 5 1 0 2 3 1 3](https://media.kunduz.com/media/sug-question-candidate/20231206024146523153-4729556.jpg?w=256)
Calculus
Application of derivativesFind the function of the form y log x whose graph is given y II 1 0 0 5 0 5 1 0 2 3 1 3
![Use the Laws of Logarithms to expand the expression log V 7x y HODINY FR](https://media.kunduz.com/media/sug-question-candidate/20231206024225488569-4729556.jpg?w=256)
Calculus
Application of derivativesUse the Laws of Logarithms to expand the expression log V 7x y HODINY FR
![Use the elimination method to solve the following initial value problem dR dt dJ dt J R R 0 1 J 0 0](https://media.kunduz.com/media/sug-question-candidate/20231206022012952806-4641188.jpg?w=256)
Calculus
Application of derivativesUse the elimination method to solve the following initial value problem dR dt dJ dt J R R 0 1 J 0 0
![Find the sum 7 1 0](https://media.kunduz.com/media/sug-question-candidate/20231206014855583409-5836419.jpg?w=256)
![A weight is oscillating on the end of a spring see figure The position of the weight relative to the point of equilibrium is given by y cos 8t 3 sin 8t where y is the displacement in meters and it is the time in seconds Find the times at which the weight is at the point of equilibrium y 0 for 0 sts 1 Round your answers to two decimal p Enter your answers as a comma separated list t H Equilibrium 4x4](https://media.kunduz.com/media/sug-question-candidate/20231206001512956315-5836419.jpg?w=256)
Calculus
Application of derivativesA weight is oscillating on the end of a spring see figure The position of the weight relative to the point of equilibrium is given by y cos 8t 3 sin 8t where y is the displacement in meters and it is the time in seconds Find the times at which the weight is at the point of equilibrium y 0 for 0 sts 1 Round your answers to two decimal p Enter your answers as a comma separated list t H Equilibrium 4x4
![The length of each of the two equal sides of an isosceles triangle is 10 meters see figure The angle between the two sides is 0 10 m a Write the area of the triangle as a function of 0 2 A 0 2 0 10 m b Write the area of the triangle as a function of and determine the value of 0 such that the area is a maximum A 0 radians](https://media.kunduz.com/media/sug-question-candidate/20231206001823437413-5836419.jpg?w=256)
Calculus
Application of derivativesThe length of each of the two equal sides of an isosceles triangle is 10 meters see figure The angle between the two sides is 0 10 m a Write the area of the triangle as a function of 0 2 A 0 2 0 10 m b Write the area of the triangle as a function of and determine the value of 0 such that the area is a maximum A 0 radians
![Rewrite the expression in terms of the first power of the cosine Use a graphing utility to graph both expressions to verify that both forms are the same 3cos 2x](https://media.kunduz.com/media/sug-question-candidate/20231206001637697589-5836419.jpg?w=256)
Calculus
Application of derivativesRewrite the expression in terms of the first power of the cosine Use a graphing utility to graph both expressions to verify that both forms are the same 3cos 2x
![Use the product to sum formulas to write the product as a sum or difference 2 sin 20 sin 50](https://media.kunduz.com/media/sug-question-candidate/20231206001752714642-5836419.jpg?w=256)
Calculus
Application of derivativesUse the product to sum formulas to write the product as a sum or difference 2 sin 20 sin 50
![Use inverse functions where necessary to solve the equation Use n as an integer constant Enter your answers as a comma separated list 2 cos x 13 sin x 8](https://media.kunduz.com/media/sug-question-candidate/20231206001446834704-5836419.jpg?w=256)
Calculus
Application of derivativesUse inverse functions where necessary to solve the equation Use n as an integer constant Enter your answers as a comma separated list 2 cos x 13 sin x 8
![Use the properties of logarithms to verify the equation In 18 2 In 3 In 2 In 18 In 32 2 In 32 2 2 In 3 In 2 X In 3 X](https://media.kunduz.com/media/sug-question-candidate/20231205225656227513-5836419.jpg?w=256)
Calculus
Application of derivativesUse the properties of logarithms to verify the equation In 18 2 In 3 In 2 In 18 In 32 2 In 32 2 2 In 3 In 2 X In 3 X
![The relationship between the number of decibels and the intensity of a sound S in watts per square meter is given by 10910 10 12 B 10 log10 a Use the properties of logarithms to write the formula in a simpler form B b Use a graphing utility to complete the table Verify your answers algebraically 10 4 10 6 10 8 S n X DO so Po 100 10 10 n 10 12 an 10 14](https://media.kunduz.com/media/sug-question-candidate/20231205225907518283-5836419.jpg?w=256)
Calculus
Application of derivativesThe relationship between the number of decibels and the intensity of a sound S in watts per square meter is given by 10910 10 12 B 10 log10 a Use the properties of logarithms to write the formula in a simpler form B b Use a graphing utility to complete the table Verify your answers algebraically 10 4 10 6 10 8 S n X DO so Po 100 10 10 n 10 12 an 10 14
![Use the properties of logarithms to expand the expression as a sum difference and or constant multiple of logarithms Assume all variables are positive](https://media.kunduz.com/media/sug-question-candidate/20231205225718067615-5836419.jpg?w=256)
Calculus
Application of derivativesUse the properties of logarithms to expand the expression as a sum difference and or constant multiple of logarithms Assume all variables are positive
![Rewrite the logarithm as a ratio of common logarithms and natural logarithms 9 lage 0 10 a common logarithms k 9 log 10 log a b natural logarithms](https://media.kunduz.com/media/sug-question-candidate/20231205225540023290-5836419.jpg?w=256)
Calculus
Application of derivativesRewrite the logarithm as a ratio of common logarithms and natural logarithms 9 lage 0 10 a common logarithms k 9 log 10 log a b natural logarithms
![Rewrite the expression in terms of In 2 and In 3 In 108](https://media.kunduz.com/media/sug-question-candidate/20231205225629024122-5836419.jpg?w=256)
![Consider the function f x x e6 f x has two inflection points at x C and x D with C D 3 6 where Cis 3 and Dis 3 6 3 Add Work X X retry this question below Finally for each of the following intervals tell whether f x is concave up type in CU or concave down type in CD C C D D](https://media.kunduz.com/media/sug-question-candidate/20231205201834964986-5868017.jpg?w=256)
Calculus
Application of derivativesConsider the function f x x e6 f x has two inflection points at x C and x D with C D 3 6 where Cis 3 and Dis 3 6 3 Add Work X X retry this question below Finally for each of the following intervals tell whether f x is concave up type in CU or concave down type in CD C C D D
![Complete the table to determine the balance A for 10 000 invested at rate r 2 for t years compounded continuously Round your answers to two decimal places 1 t 1 10 20 30 40 50](https://media.kunduz.com/media/sug-question-candidate/20231205195541458745-5836419.jpg?w=256)
Calculus
Application of derivativesComplete the table to determine the balance A for 10 000 invested at rate r 2 for t years compounded continuously Round your answers to two decimal places 1 t 1 10 20 30 40 50
![Find the x and y intercepts of the graph of the circle If an answer does not exist enter DNE x 10x y 2y 19 0 STEP 1 Consider the equation for the circle x2 10x y2 2y 19 0 Let y 0 Give the resulting equation below 10x 19 0 STEP 2 Solve your equation from Step 1 for x If one or more solutions exist these values are the x coordinates of the x intercepts of the graph of the circle Enter your answers as a comma separated list x x STEP 3 Consider once again the equation for the circle x 10x y 2y 190 Now let x 0 Give the resulting equation below 2y 19 0 STEP 4 Solve your equation from Step 3 for y If one or more solutions exist these values are the y coordinates of the y intercepts of the graph of the circle Enter your answers as a comma separated list y x STEP 5 Complete the problem by listing the x and y intercepts x intercepts x y y intercept x y x y x x smaller x value larger x value](https://media.kunduz.com/media/sug-question-candidate/20231205185118250655-5836419.jpg?w=256)
Calculus
Application of derivativesFind the x and y intercepts of the graph of the circle If an answer does not exist enter DNE x 10x y 2y 19 0 STEP 1 Consider the equation for the circle x2 10x y2 2y 19 0 Let y 0 Give the resulting equation below 10x 19 0 STEP 2 Solve your equation from Step 1 for x If one or more solutions exist these values are the x coordinates of the x intercepts of the graph of the circle Enter your answers as a comma separated list x x STEP 3 Consider once again the equation for the circle x 10x y 2y 190 Now let x 0 Give the resulting equation below 2y 19 0 STEP 4 Solve your equation from Step 3 for y If one or more solutions exist these values are the y coordinates of the y intercepts of the graph of the circle Enter your answers as a comma separated list y x STEP 5 Complete the problem by listing the x and y intercepts x intercepts x y y intercept x y x y x x smaller x value larger x value
![1 If x 0 and y 0 then 2x y al X](https://media.kunduz.com/media/sug-question-candidate/20231205103936810862-6333882.jpg?w=256)
![Graph the function not by plotting points but by starting from the graph of y e in the figure below 10 range O 10 O y 10 10 y 10 10 y 2 10 10 X 10 O 10 State the domain and range Enter your answers using interval notation domain y 10 10 10 10](https://media.kunduz.com/media/sug-question-candidate/20231205030205618345-4729556.jpg?w=256)
Calculus
Application of derivativesGraph the function not by plotting points but by starting from the graph of y e in the figure below 10 range O 10 O y 10 10 y 10 10 y 2 10 10 X 10 O 10 State the domain and range Enter your answers using interval notation domain y 10 10 10 10
![Use transformations of the graph of y 1 x to graph the rational function as in Example 2 domain 10 range 10 5 5 10 X 10 X 10 State the domain and range Enter your answers using interval notation 10 5 5 2](https://media.kunduz.com/media/sug-question-candidate/20231205024404727243-4729556.jpg?w=256)
Calculus
Application of derivativesUse transformations of the graph of y 1 x to graph the rational function as in Example 2 domain 10 range 10 5 5 10 X 10 X 10 State the domain and range Enter your answers using interval notation 10 5 5 2
![150 180 210 120 240 90 270 60 30 2 3 4 300 330 0](https://media.kunduz.com/media/sug-question-candidate/20221128210245146787-4338747.jpg?w=256)
![Solve the triangle with the given angles and sides 6 points B 52 C 28 a 43 c](https://media.kunduz.com/media/sug-question-candidate/20231204224410307209-5947623.jpg?w=256)
Calculus
Application of derivativesSolve the triangle with the given angles and sides 6 points B 52 C 28 a 43 c
![Find the area of the parallelogram whose vertices are given below A 1 4 B 4 0 C 11 3 and D 6 7](https://media.kunduz.com/media/sug-question-candidate/20231204221643495773-4724413.jpg?w=256)
Calculus
Application of derivativesFind the area of the parallelogram whose vertices are given below A 1 4 B 4 0 C 11 3 and D 6 7
![Find the volume of the parallelepiped box determined by u v and w u 4i 2j V 3i j 3k w i 3k](https://media.kunduz.com/media/sug-question-candidate/20231204221658188271-4724413.jpg?w=256)
Calculus
Application of derivativesFind the volume of the parallelepiped box determined by u v and w u 4i 2j V 3i j 3k w i 3k
![0 Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of 0 where 2 3 3 81 9x x 3 cos 0 0 1 1 Then find sin 0 and](https://media.kunduz.com/media/sug-question-candidate/20231204203424797788-6332946.jpg?w=256)
Calculus
Application of derivatives0 Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of 0 where 2 3 3 81 9x x 3 cos 0 0 1 1 Then find sin 0 and
![The price of a home is 180 000 The mortgage on the hous a 30 year fixed rate mortgage at 4 Use this information to fin the amount of the monthly payment Round your answer to the nearest cent Use the formula below to solve the problem P 1 1 xt PMT 859 35 954 83 012 54](https://media.kunduz.com/media/sug-question-candidate/20231204200952790679-6194332.jpg?w=256)
Calculus
Application of derivativesThe price of a home is 180 000 The mortgage on the hous a 30 year fixed rate mortgage at 4 Use this information to fin the amount of the monthly payment Round your answer to the nearest cent Use the formula below to solve the problem P 1 1 xt PMT 859 35 954 83 012 54
![18 Estimate to the hundredth the area from 1 to 5 under the graph of f f x approximating rectangles and right endpoints 2 x using four](https://media.kunduz.com/media/sug-question-candidate/20231204192909278146-5994856.jpg?w=256)
Calculus
Application of derivatives18 Estimate to the hundredth the area from 1 to 5 under the graph of f f x approximating rectangles and right endpoints 2 x using four
![The population of a colony of mosquitoes obeys the law of uninhibited growth If there are 1000 mosquitoes initially and there are 1200 after 1 day what is a The growth rate Round to three decimal places b The population of mosquitoes after 6 days Round to the nearest whole number c When will the population double Round to the nearest whole number](https://media.kunduz.com/media/sug-question-candidate/20231204190103623225-5472891.jpg?w=256)
Calculus
Application of derivativesThe population of a colony of mosquitoes obeys the law of uninhibited growth If there are 1000 mosquitoes initially and there are 1200 after 1 day what is a The growth rate Round to three decimal places b The population of mosquitoes after 6 days Round to the nearest whole number c When will the population double Round to the nearest whole number
![Given that 2 is a zero find the remaining real zeros Show work WILL be required on the exam No approximate answers will be accepted 3 x 4x 3x 2 Solution 1 Solution 2](https://media.kunduz.com/media/sug-question-candidate/20231204182357383419-5472891.jpg?w=256)
Calculus
Application of derivativesGiven that 2 is a zero find the remaining real zeros Show work WILL be required on the exam No approximate answers will be accepted 3 x 4x 3x 2 Solution 1 Solution 2
![5 10 5 5 5 10 For which values in the domain is this graph increasing Give your answer in interval notation Assume the graph extends to finity in both directions Submit](https://media.kunduz.com/media/sug-question-candidate/20231204181948374949-5472891.jpg?w=256)
Calculus
Application of derivatives5 10 5 5 5 10 For which values in the domain is this graph increasing Give your answer in interval notation Assume the graph extends to finity in both directions Submit
![Find the solution to the equation given below Exact values only Try again 4x 5 x 0 4024](https://media.kunduz.com/media/sug-question-candidate/20231204175520066785-5472891.jpg?w=256)
Calculus
Application of derivativesFind the solution to the equation given below Exact values only Try again 4x 5 x 0 4024
![What is the conjugate of b if a 3 4i and b 2 3i](https://media.kunduz.com/media/sug-question-candidate/20231204180410671594-5472891.jpg?w=256)
![Question What is a b if a 3 4i and b 2 3i](https://media.kunduz.com/media/sug-question-candidate/20231204180234193841-5472891.jpg?w=256)