Application of derivatives Questions and Answers
![Find the center vertices foci and asymptotes of the hyperbola x 9 x 1 25 Center 4 Acv 1 Vertices and 0 Foci and D and y 0 O](https://media.kunduz.com/media/sug-question-candidate/20230626024036251962-5366129.jpg?w=256)
Calculus
Application of derivativesFind the center vertices foci and asymptotes of the hyperbola x 9 x 1 25 Center 4 Acv 1 Vertices and 0 Foci and D and y 0 O
![Use term by term differentiation or integration to find a power series centered at x 0 for 6x5 1 x6 f x](https://media.kunduz.com/media/sug-question-candidate/20230626012730763246-3636182.jpg?w=256)
Calculus
Application of derivativesUse term by term differentiation or integration to find a power series centered at x 0 for 6x5 1 x6 f x
![The graph of a radical function f x is shown below y f x k a x h then the product of a h k](https://media.kunduz.com/media/sug-question-candidate/20230626000515467125-5755691.jpg?w=256)
Calculus
Application of derivativesThe graph of a radical function f x is shown below y f x k a x h then the product of a h k
![Assume the acceleration of the object is a t 32 feet per second per second Neglect air resistance With what initial velocity must an object be thrown upward from ground level to reach the top of a national monument 510 feet Round your answer to three decimal places ft sec](https://media.kunduz.com/media/sug-question-candidate/20230625201452122790-3917327.jpg?w=256)
Calculus
Application of derivativesAssume the acceleration of the object is a t 32 feet per second per second Neglect air resistance With what initial velocity must an object be thrown upward from ground level to reach the top of a national monument 510 feet Round your answer to three decimal places ft sec
![For the given functions a express 2x w 6x 3y x cost y sint t 3 dw a Express as a function of t dt dw dw dw as a function of t both by using the Chain Rule and by expressing w in terms of t and differentiating directly with respect to t Then b evaluate at the given value dt dt dt](https://media.kunduz.com/media/sug-question-candidate/20230625200538330669-4709559.jpg?w=256)
Calculus
Application of derivativesFor the given functions a express 2x w 6x 3y x cost y sint t 3 dw a Express as a function of t dt dw dw dw as a function of t both by using the Chain Rule and by expressing w in terms of t and differentiating directly with respect to t Then b evaluate at the given value dt dt dt
![dz se the functions below to a express and dz du as functions of u and v both by using the Chain Rule and by expressing z directly in terms of u and v before differentiating Then b evaluate and dv z 2 tan 1 z dz a Express and as functions of x and y Find dx dz dy dz x u cos v yusin v u v 0 8 ax Type an exact answer dz du dz av at the given point C](https://media.kunduz.com/media/sug-question-candidate/20230625200650469402-4709559.jpg?w=256)
Calculus
Application of derivativesdz se the functions below to a express and dz du as functions of u and v both by using the Chain Rule and by expressing z directly in terms of u and v before differentiating Then b evaluate and dv z 2 tan 1 z dz a Express and as functions of x and y Find dx dz dy dz x u cos v yusin v u v 0 8 ax Type an exact answer dz du dz av at the given point C
![3 You can either row your boat from point A to point C Or you can row from point A to some point P between B and C and then run the rest of the way to point C If you row at an average rate of 3 km h and run at an average rate of 5 km h what is the shortest amount of time it will take you to get to point C 15 points 5 km](https://media.kunduz.com/media/sug-question-candidate/20230625131309804087-5756279.jpg?w=256)
Calculus
Application of derivatives3 You can either row your boat from point A to point C Or you can row from point A to some point P between B and C and then run the rest of the way to point C If you row at an average rate of 3 km h and run at an average rate of 5 km h what is the shortest amount of time it will take you to get to point C 15 points 5 km
![How much work is done in J when a weight lifter lifts 130 kg from 1 5 m to 2 0 m above the ground Use 9 8 m s for g J](https://media.kunduz.com/media/sug-question-candidate/20230625044909557181-4426144.jpg?w=256)
Calculus
Application of derivativesHow much work is done in J when a weight lifter lifts 130 kg from 1 5 m to 2 0 m above the ground Use 9 8 m s for g J
![Compute the work done in ft lb in hoisting a 1 500 lb grand piano from the ground up to the third floor 30 feet above the ground ft lb](https://media.kunduz.com/media/sug-question-candidate/20230625044922742424-4426144.jpg?w=256)
Calculus
Application of derivativesCompute the work done in ft lb in hoisting a 1 500 lb grand piano from the ground up to the third floor 30 feet above the ground ft lb
![Your brother has the flu At 6 p m you checked his temperature and it was 100 1 F You then check it at 7 PM and it has increased to 101 F You know a dangerous temperature is anything above 103 a If his temperature continues to increase at this rate what time should you take him to the hospital Create a table of values to justify your conclusion it doesn t have to be pretty just make sure it s clear what s going on b Use your table to calculate the slope and write an equation in slope intercept form c What does the slope mean in this problem d What does the y intercept represent Does it have meaning in this problem e Use your equation to graph a line Mark the place on your graph time when your brother s temperature is critical What are the coordinates of this point f Should you go to the emergency room now or can you wait until tomorrow morning g Using this equation determine what time his](https://media.kunduz.com/media/sug-question-candidate/20230625024218042139-5491399.jpg?w=256)
Calculus
Application of derivativesYour brother has the flu At 6 p m you checked his temperature and it was 100 1 F You then check it at 7 PM and it has increased to 101 F You know a dangerous temperature is anything above 103 a If his temperature continues to increase at this rate what time should you take him to the hospital Create a table of values to justify your conclusion it doesn t have to be pretty just make sure it s clear what s going on b Use your table to calculate the slope and write an equation in slope intercept form c What does the slope mean in this problem d What does the y intercept represent Does it have meaning in this problem e Use your equation to graph a line Mark the place on your graph time when your brother s temperature is critical What are the coordinates of this point f Should you go to the emergency room now or can you wait until tomorrow morning g Using this equation determine what time his
![Among all rectangles that have a perimeter of 64 find the dimensions of the one whose area is largest Write your answers as fractions reduced to lowest terms Answer Keypad Keyboard Shortcuts](https://media.kunduz.com/media/sug-question-candidate/20230625005111787557-5485104.jpg?w=256)
Calculus
Application of derivativesAmong all rectangles that have a perimeter of 64 find the dimensions of the one whose area is largest Write your answers as fractions reduced to lowest terms Answer Keypad Keyboard Shortcuts
![A projectile is fired with an initial speed of 220 m s and angle of elevation 60 The projectile is fired from a position 90 m above the ground Use g 9 8 m s2 Round your answers to the nearest whole number a Find the range in m of the projectile 4276 7 x m b Find the maximum height in m reached 1942 m c Find the speed in m s at impact 224 m s](https://media.kunduz.com/media/sug-question-candidate/20230624224907484502-3742842.jpg?w=256)
Calculus
Application of derivativesA projectile is fired with an initial speed of 220 m s and angle of elevation 60 The projectile is fired from a position 90 m above the ground Use g 9 8 m s2 Round your answers to the nearest whole number a Find the range in m of the projectile 4276 7 x m b Find the maximum height in m reached 1942 m c Find the speed in m s at impact 224 m s
![Tutorial Exercise The position function of a particle is given by r t t 4t t 16t When is the speed a minimum Step 1 To find when the speed is a minimum we need to find the speed as a function of t then find its derivative and see when it v t r t 2t 4 2t 16 2t 4 2t 16 Step 2 We remember that the speed is the magnitude of the velocity vector and is calculated as follows Iv t 2t 4 2t 16 V 8 Step 3 Next we use the Chain Rule to find the derivative of the speed v t 8 64t 272 1 2 8t 32 X 2 64 t 272 272](https://media.kunduz.com/media/sug-question-candidate/20230624223321213783-3742842.jpg?w=256)
Calculus
Application of derivativesTutorial Exercise The position function of a particle is given by r t t 4t t 16t When is the speed a minimum Step 1 To find when the speed is a minimum we need to find the speed as a function of t then find its derivative and see when it v t r t 2t 4 2t 16 2t 4 2t 16 Step 2 We remember that the speed is the magnitude of the velocity vector and is calculated as follows Iv t 2t 4 2t 16 V 8 Step 3 Next we use the Chain Rule to find the derivative of the speed v t 8 64t 272 1 2 8t 32 X 2 64 t 272 272
![A gun has muzzle speed 158 m s Find two angles of elevation in degrees that can be used to hit a target 800 m away Enter your answers as a comma separated list Round your answers to one decimal place Use g 9 8 m s 30 4 59 6 a X 0](https://media.kunduz.com/media/sug-question-candidate/20230624210220130054-3742842.jpg?w=256)
Calculus
Application of derivativesA gun has muzzle speed 158 m s Find two angles of elevation in degrees that can be used to hit a target 800 m away Enter your answers as a comma separated list Round your answers to one decimal place Use g 9 8 m s 30 4 59 6 a X 0
![Find the total area of the shaded regions 10 164 Ay f x 64 1 X 10](https://media.kunduz.com/media/sug-question-candidate/20230624191148703339-5686493.jpg?w=256)
![Use the Substitution Formula to evaluate the integral 0 1 8 7 1 0 1 1 1 18 7 dt](https://media.kunduz.com/media/sug-question-candidate/20230624191936421114-5686493.jpg?w=256)
Calculus
Application of derivativesUse the Substitution Formula to evaluate the integral 0 1 8 7 1 0 1 1 1 18 7 dt
![What is the total resistance for a circuit with three resistors in series if the resistances are 2 ohms 5 ohms and 4 ohms respectively Draw a diagram of these resistors in series What is the total resistance for a circuit with two parallel resistors one with a resistance of 3 ohms and the other with a resistance of 7 ohms 4 Draw a diagram of these resistors in parallel 5 Consider a circuit with two parallel resistors a Solve the total resistance equation for Rr Also express it as one fraction](https://media.kunduz.com/media/sug-question-candidate/20230624184746673782-5066220.jpg?w=256)
Calculus
Application of derivativesWhat is the total resistance for a circuit with three resistors in series if the resistances are 2 ohms 5 ohms and 4 ohms respectively Draw a diagram of these resistors in series What is the total resistance for a circuit with two parallel resistors one with a resistance of 3 ohms and the other with a resistance of 7 ohms 4 Draw a diagram of these resistors in parallel 5 Consider a circuit with two parallel resistors a Solve the total resistance equation for Rr Also express it as one fraction
![Evaluate the integral 4e4t S4e 2 e4t 4e4t dt](https://media.kunduz.com/media/sug-question-candidate/20230624191758649738-5686493.jpg?w=256)
![Use substitution to find the given indefinite integral Sx 9x 7 dx 2](https://media.kunduz.com/media/sug-question-candidate/20230624191702009332-5686493.jpg?w=256)
Calculus
Application of derivativesUse substitution to find the given indefinite integral Sx 9x 7 dx 2
![4 Find the sum given below 10 51 3 4](https://media.kunduz.com/media/sug-question-candidate/20230624170452986327-3756363.jpg?w=256)
![11 Solve the differential equation df 16u 3 du 8u 3u 2](https://media.kunduz.com/media/sug-question-candidate/20230624170710988936-3756363.jpg?w=256)
![function is given below Determine the average rate of change of the function between x 5 and x 2 h x x 2 x](https://media.kunduz.com/media/sug-question-candidate/20230624170646368048-5755317.jpg?w=256)
Calculus
Application of derivativesfunction is given below Determine the average rate of change of the function between x 5 and x 2 h x x 2 x
![Find F x given x 1 F x f 2z 5 dz](https://media.kunduz.com/media/sug-question-candidate/20230624170559309457-3756363.jpg?w=256)
![Find the equation of the line tangent to the graph of y sin x at x 0](https://media.kunduz.com/media/sug-question-candidate/20230624155325772814-5755317.jpg?w=256)
Calculus
Application of derivativesFind the equation of the line tangent to the graph of y sin x at x 0
![u 7 du Describe the most appropriate substitution case and the values of u and du Select the correct choice below and fill in the answer boxes within your choi A Substitute u for the quantity in the denominator Let v so that dv du B Substitute u for the quantity under the root Let v u 7 so that dv 1 du C Substitute u for the quantity in the numerator Let v so that dv du Use the substitution to evaluate the integral](https://media.kunduz.com/media/sug-question-candidate/20230624095756834621-5686493.jpg?w=256)
Calculus
Application of derivativesu 7 du Describe the most appropriate substitution case and the values of u and du Select the correct choice below and fill in the answer boxes within your choi A Substitute u for the quantity in the denominator Let v so that dv du B Substitute u for the quantity under the root Let v u 7 so that dv 1 du C Substitute u for the quantity in the numerator Let v so that dv du Use the substitution to evaluate the integral
![For the given functions a express as a function of t both by using the Chain Rule and by expressing w in terms dw of t and differentiating directly with respect to t Then b evaluate at the given value of t dt w 7yex Inz x In 1 y tant z e5t t 1 a b dw dt dw 14t tan dt It 1 dw dt t 2 Type an exact answer CODE](https://media.kunduz.com/media/sug-question-candidate/20230624093516963193-4710940.jpg?w=256)
Calculus
Application of derivativesFor the given functions a express as a function of t both by using the Chain Rule and by expressing w in terms dw of t and differentiating directly with respect to t Then b evaluate at the given value of t dt w 7yex Inz x In 1 y tant z e5t t 1 a b dw dt dw 14t tan dt It 1 dw dt t 2 Type an exact answer CODE
![Find the total area of the shaded regions 16 Q Q The total area of the shad Type an integer or a simp](https://media.kunduz.com/media/sug-question-candidate/20230624081100702700-5686493.jpg?w=256)
Calculus
Application of derivativesFind the total area of the shaded regions 16 Q Q The total area of the shad Type an integer or a simp
![se the Substitution Formula to evaluate the integral 0 1 4 14 dt Simplify your answer 0 S 4 14 d dt 1](https://media.kunduz.com/media/sug-question-candidate/20230624081630104982-5686493.jpg?w=256)
Calculus
Application of derivativesse the Substitution Formula to evaluate the integral 0 1 4 14 dt Simplify your answer 0 S 4 14 d dt 1
![Use substitution to find the indefinite integral 3x x dx 8x x 8x](https://media.kunduz.com/media/sug-question-candidate/20230624072505094859-5686493.jpg?w=256)
![Evaluate the expression 14 7 8x2 9 14 7 8x2 9 Simplify your answer](https://media.kunduz.com/media/sug-question-candidate/20230624072226132556-5366129.jpg?w=256)
Calculus
Application of derivativesEvaluate the expression 14 7 8x2 9 14 7 8x2 9 Simplify your answer
![1 D 1 2 5 6 3 4 Graph of f The figure above shows the graph of a function fi which has a vertical tengont at x 2 and a horizontal tangent at X 4 Which of the foroving statements is fabe fis not differentiable at x 1 because the graph of thes Jump discontinuity at x 1 fs not differentiable at X 2 because the graph of has a vertical tergent et X 2 Gf is not diffentiable at x 6 because the graph off hes I removable discontinuity e x b Of is not differentiable 24 x 4 because the graph of f has a horizonta tangent at X 4](https://media.kunduz.com/media/sug-question-candidate/20230624071423626642-4832772.jpg?w=256)
Calculus
Application of derivatives1 D 1 2 5 6 3 4 Graph of f The figure above shows the graph of a function fi which has a vertical tengont at x 2 and a horizontal tangent at X 4 Which of the foroving statements is fabe fis not differentiable at x 1 because the graph of thes Jump discontinuity at x 1 fs not differentiable at X 2 because the graph of has a vertical tergent et X 2 Gf is not diffentiable at x 6 because the graph off hes I removable discontinuity e x b Of is not differentiable 24 x 4 because the graph of f has a horizonta tangent at X 4
![Let F x x 81 Ix 91 Find the followu following Tim X 79 F X lim F x X 9 18 and 9 618 and 18 G both 18 limits a](https://media.kunduz.com/media/sug-question-candidate/20230624065314361837-4832772.jpg?w=256)
Calculus
Application of derivativesLet F x x 81 Ix 91 Find the followu following Tim X 79 F X lim F x X 9 18 and 9 618 and 18 G both 18 limits a
![use continuity to evaluate the limit Sin x 4 5in x lim X 7137 6 13T 1](https://media.kunduz.com/media/sug-question-candidate/20230624064219473442-4832772.jpg?w=256)
Calculus
Application of derivativesuse continuity to evaluate the limit Sin x 4 5in x lim X 7137 6 13T 1
![Find the lim lim x 2x X 2 X 4x 4 X 2 7 2](https://media.kunduz.com/media/sug-question-candidate/20230624062754775990-4832772.jpg?w=256)
![Find the limit lim x 3 x x 6 X Select the correct ans 200 1 2](https://media.kunduz.com/media/sug-question-candidate/20230624060929480395-4832772.jpg?w=256)
![Find the given derivative by finding the first few derivatives and observing the pattern that occurs sin x d 30 dx 30](https://media.kunduz.com/media/sug-question-candidate/20230624053229628897-4832772.jpg?w=256)
Calculus
Application of derivativesFind the given derivative by finding the first few derivatives and observing the pattern that occurs sin x d 30 dx 30
![ind d 2020of where for xex dy 2020 dx](https://media.kunduz.com/media/sug-question-candidate/20230624053807693046-4832772.jpg?w=256)
![Determine where of is discontinuos if x 0 if 0 x 3 x 3 f x 2 x X 3 X 3 X if](https://media.kunduz.com/media/sug-question-candidate/20230624053251446582-4832772.jpg?w=256)
Calculus
Application of derivativesDetermine where of is discontinuos if x 0 if 0 x 3 x 3 f x 2 x X 3 X 3 X if
![Evaluate the integral 2 2 X 0x X 2 dx 2 x 4x 7 X 2 4x 7 dx](https://media.kunduz.com/media/sug-question-candidate/20230624043307056146-5686493.jpg?w=256)
![Draw a graph to match the description given G x is decreasing over 0 0 and 7 00 and increasing over 0 7](https://media.kunduz.com/media/sug-question-candidate/20230624040147068056-5754622.jpg?w=256)
Calculus
Application of derivativesDraw a graph to match the description given G x is decreasing over 0 0 and 7 00 and increasing over 0 7
![Use the Substitution Formula to evaluate the integral 3 413 0 4 14 dt 2](https://media.kunduz.com/media/sug-question-candidate/20230624043213857154-5686493.jpg?w=256)
Calculus
Application of derivativesUse the Substitution Formula to evaluate the integral 3 413 0 4 14 dt 2
![Evaluate the definite integral 9 10y y 24 y dy](https://media.kunduz.com/media/sug-question-candidate/20230624043108689524-5686493.jpg?w=256)
![Find the indefinite integral fin s In 9x 8 dx C](https://media.kunduz.com/media/sug-question-candidate/20230624041927189336-5686493.jpg?w=256)
![The rate of growth of a population in millions per year in the year x is approximated by f x xe a Find the total population function Remember that the population is given in millions b As time goes on what will happen to the total population a Set up the appropriate integral that can be used to find the population function The total population in the fourth year is 70 000](https://media.kunduz.com/media/sug-question-candidate/20230624030950093900-5686493.jpg?w=256)
Calculus
Application of derivativesThe rate of growth of a population in millions per year in the year x is approximated by f x xe a Find the total population function Remember that the population is given in millions b As time goes on what will happen to the total population a Set up the appropriate integral that can be used to find the population function The total population in the fourth year is 70 000
![Find the given indefinite integral State whether integration by substitution or integration by parts was us f5x 9dx Using integration by Type an exact answ 5x 9 substitution dx](https://media.kunduz.com/media/sug-question-candidate/20230624031143556783-5686493.jpg?w=256)
Calculus
Application of derivativesFind the given indefinite integral State whether integration by substitution or integration by parts was us f5x 9dx Using integration by Type an exact answ 5x 9 substitution dx
![Use substitution to find the indefinite integral 1 1 301 151 d t e dt Describe the most appropriate substitution case and the values of u and du Select the correct choice below and fill in the answer boxes within ye OA Substitute u for the quantity under a root or raised to a power Let u so that du dt so that du OB Substitute u for the exponent on e Let u OC Substitute u for the quantity in the denominator Let u dt so that du dt](https://media.kunduz.com/media/sug-question-candidate/20230624030652300328-5686493.jpg?w=256)
Calculus
Application of derivativesUse substitution to find the indefinite integral 1 1 301 151 d t e dt Describe the most appropriate substitution case and the values of u and du Select the correct choice below and fill in the answer boxes within ye OA Substitute u for the quantity under a root or raised to a power Let u so that du dt so that du OB Substitute u for the exponent on e Let u OC Substitute u for the quantity in the denominator Let u dt so that du dt
![Evaluate the integral e t Se 7 t t dt](https://media.kunduz.com/media/sug-question-candidate/20230624030714502271-5686493.jpg?w=256)
![Use substitution to find the indefinite integral f8 8x 8x 11 dx](https://media.kunduz.com/media/sug-question-candidate/20230624030519228531-5686493.jpg?w=256)
![What is f x given the integral function f x cos e dur o f x 2x cos ex o f x x cos ex o f x 2x cos ex o f x x cos ex 35 36 37 38 39 40](https://media.kunduz.com/media/sug-question-candidate/20220622012630575980-3893386.jpg?w=256)
Calculus
Application of derivativesWhat is f x given the integral function f x cos e dur o f x 2x cos ex o f x x cos ex o f x 2x cos ex o f x x cos ex 35 36 37 38 39 40
![Polar equation r 2sin 30 Outside of the unit circle r 1 See graph Find the area of the shaded part 3](https://media.kunduz.com/media/sug-question-candidate/20220622003609437534-4186878.jpg?w=256)
Calculus
Application of derivativesPolar equation r 2sin 30 Outside of the unit circle r 1 See graph Find the area of the shaded part 3