Calculus Questions

A cylindrical tank with radius 7 m is being filled with water at a rate of 4 m min How fast is the height of the water increasing in m min m min

Use differentials to estimate the amount of paint needed in m to apply a coat of paint 0 06 cm thick to a hemispherical don with diameter 54 m Round your answer to two decimal places m

1 If two resistors with resistances R and R are connected in parallel as shown in the figure then the total resistance R measured in ohms 22 is given by R R 22 s www R 150 1 R 1 R If R and R are increasing at rates of 0 3 s and 0 2 n s respectively how fast in s is R changing when R 90 and R 100 2 Round your answer to three decimal places

Find the differential of each function a y x sin 4x dy b y In 2 1 dy

A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m highe than the bow of the boat If the rope is pulled in at a rate of 1 m s how fast in m s is the boat approaching the dock when it is 7 m from the dock Round your answer to two decimal places m s

Two sides of a triangle have lengths 11 m and 17 m The angle between them is increasing at a rate of 2 min How fast in m min is the length of the third side increasing when the angle between the sides of fixed length is 60 Hint Use the Law of Cosines Round your answer to three decimal places m min

Find by implicit differentiation dy dx dy dx II tan 5x y x 4xy 10yx 100y x 25y x 1 x 200x 8y 5x 43 Need Help Submit Answer 1 1 Points Read It DETAILS Watch It X

Boyle s Law states that when a sample of gas is compressed at a constant temperature the pressure P and volume V satisfy the equation PV C where C is a constant Suppose that at a certain instant the volume is 300 cm the pressure is 120 kPa and the pressure is increasing at a rate of 20 kPa min At what rate in cm min is the volume decreasing at this instant cm min

Find the linearization L x of the function at a f x x a 4 L x

The altitude of a triangle is increasing at a rate of 1 cm min while the area of the triangle is increasing at a rate of 2 cm2 min At what rate is the base of the triangle changing when the altitude is 10 cm and the area is 190 cm cm min

The length of a rectangle is increasing at a rate of 3 cm s and its width is increasing at a rate of 8 cm s When the length is 7 cm and the width is 5 cm how fast is the area of the rectangle increasing in cm s cm s

Two sides of a triangle are 8 m and 9 m in length and the angle between them is increasing at a rate of 0 06 rad s Find the rate in m s at which the area of the triangle is increasing when the angle between the sides of fixed length is rad 3 m s

Each side of a square is increasing at a rate of 5 cm s At what rate in cm s is the area of the square increasing when the area of the square is 81 cm cm s

A kite 100 ft above the ground moves horizontally at a speed of 4 ft s At what rate in rad s is the angle in radians between the string and the horizontal decreasing when 200 ft of string have been let out rad s

Use the Limit Comparison Test to determine if the following series converges or diverges Hint Limit Comparison with n 5 3 2 n 1 n n 8 1 The series n 5 and Apply the Limit Comparison Test with an 2 bn Complete the sentence below n 1 n an n 1 n bn n 5 3 2 n 1 n n 8 because lim an bn n n an b and for all

Determine if the series converges or diverges Give a reason for your answer 2n 3n 7 n 1 Select the correct choice below and if necessary fill in the answer box to complete your choice V The series converges because the limit found in the nth Term Test is OA Type an integer or a fraction The series diverges because the limit found in the nth Term Test is B O B Type an integer or a fraction The Direct Comparison Test with En shows that the series diverges O C D The Direct Comparison Test with shows that the series converges n

10 10 5 5 5 Match each graph to the expression that transforms the black solid graph into the blue dashed graph 1 5 5 0 5 0 1 1 7 1 1 1 5 5 5 5 10 10 10 10 x 2 x 2 x 1 2 x 10 10 10 10 5 5 1 5 5 0 0 5 5 10 10 10

Sketch the graph of f x y 4x 16y 4 State the domain and range of the function O A OA x y x Simplify your answer OB x y xs OC x y x OD x y x2 Simplify your answer SOB C Find the domain of the function Select the correct choice below and if necessary fill in the answer box es to complete your choice and x2 Simplify your answers Simplify your answer Use a comma to separate answers as needed O C 2 y

Abaseball pitcher s earned run average ERA is A e i 9e i where e is the number of earned runs given up by the pitcher and i is the number of innings pitched Good pitchers have low ERAS Assume that e 20 and i 0 are real numbers a During his best season a certain pitcher pitched a total of 237 innings and gave up just 36 earned runs What was his ERA b Determine the ERA of a relief pitcher who gives up 5 earned runs one third of an inning c Graph the level curve A e i 2 and describe the relationship between e and i in this case a The pitcher s ERA for the season was Round to two decimal places as needed b The relief pitcher s ERA was Round to two decimal places as needed c Graph the level curve A e i 2 Choose the correct graph below O A A 10 Q OB A 10 Describe the relationship between e and i in this case CICCESS Q O C A 10 O D A 10 Q

Drag the function equation on top of the matching graph given the transformation 6 YA 8 LL f x 2 f x 2 6 x

10 5 TT 10 5 0 5 11 X 1 VE 1 LO 5 vs 10 10 3 If the black graph is f x x then which of the following represents the dashed blue

How did you decide If f x x 5 2 then which of the following statements is true The 5 is a tight number and moves the graph horizontally to the right while the 2 is a loose number and moves the graph vertically up The 5 is a tight number and moves the graph horizontally to the left while the 2 is a loose number and moves the graph vertically up The 5 is a tight number and moves the graph horizontally to the left while the 2 is a loose number and moves the graph vertically down The 5 is a tight number and moves the graph horizontally to the right while the 2 is a loose number and moves the graph vertically down

3 Zx zy y 17 15 8x 7 y 13 0 5 6

Consider the following parametric equation What is the length of the curve for 0 0 to 0 x 13 cos 0 0 sin 0 y 13 sin 0 0 cos 0

Match the graph of the function z 2 x y to the system of the level curves Which of the following level curves matches the given surface graph OA OB X C SEM SAVAN SVE TERO mack PERGER ID OC

Consider the parametric curve given by the equations x t t 30t 26 y t 30t 37 How many units of distance are covered by the point P t x t y t between t 0 and t 10

The following parametric equations define an ellipse Al WWW Duran x 4 cos 0 Answer Compute the area enclosed by this ellipse Hint The ellipse is symmetric over the x axis Select limits for integration so that you are integrating left to right y 2 sin 0 0 0 2TT Graph the curve on a calculator Turn in a sketch of the curve

Consider the curve given by the parametric equations a Determine the point on the curve where the tangent is horizontal t x t t 147 y 3 t 147 b Determine the points t1 t2 where the tangent is vertical and t t t t Graph the curve on a calculator Turn in a sketch of the curve

If x et and y t 8 2 find an equation of the tangent in terms of x and y to the curve at 1 64 Tangent line Graph the curve on a calculator Turn in a sketch of the curve with the tangent line

show and and that seventh term 1 243 find them geometric sequence with first term 3 has two possible sums to infinity

If x 3t t and y 7 5t find an equation of the tangent in terms of x and y to the curve at t 1 Tangent line

If x 5t4 5 and y 5t t find the following derivatives as functions of t dy dx d y dx

a Find dy da dy da expressed as a function of t for the given the parametric equations d y dx2 b Find d y dx2 c Except for at the points where Concave expressed as a function of t 8 S cos t 7 sin t dy is undefined is the curve concave up or concave down Enter up or down da

g x x 27x 61 a Select the correct choice below and if necessary fill in the answer box within your choice OA The critical value s of the function is are Use a comma to separate answers as needed OB The function has no critical values b Select the correct choice below and if necessary fill in the answer box es within your choice OA The relative maximum point s is are and there are no relative minimum points Simplify your answer Type an ordered OB The relative minimum point s is are Simplify your answer Type an ordered pair using integers fractions Use a comma to separate answers as needed and there are no relative maximum points pair using integers or fractions Use a comma to separate answers as needed OC The relative minimum point s is are and the relative maximum point s is are Simplify your answers Type ordered pairs using integers or fractions Use a comma to separate answers as needed D There are no relative minimum points and there are no relative maximum points

f x 5 x x Describe any relative extrema Select the correct choice below and if necessary fill in the answer box es to within your choice and there are no relative maximum points OA The relative minimum point s is are Simplify your answer Type an ordered OB The relative minimum point s is are Simplify your answers Type ordered pair using integers or fractions Use a comma to separate answers as needed and the relative maximum point s is are pairs using integers or fractions Use a comma to separate answers as needed and there are no relative minimum points OC The relative maximum point s is are Simplify your answer Type an ordered pair using integers or fractions Use a comma to separate answers as needed OD There are no relative minimum points and there are no relative maximum points Identify any intervals over which the function is increasing or decreasing Select the correct choice below and fill in the answer box es within your choice and is not increasing anywhere or fractions for any numbers in the expression Use a comma to separate answers as needed and decreasing over the interval s OA The function f x is decreasing over the interval s Type your answer in interval notation Use integers OB The function f x is increasing over the interval s Type your answers in interval notation Use integers or fractions for any numbers in the expressions Use a comma to separate answers as needed and is not decreasing anywhere OC The function f x is increasing over the interval s Type your answer in interval notation Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed Choose the correct graph below O A Q Q G OB Ay Q O C Ay Q O D OU

Graph the polynomial function 3 2 f x x 3x 13x 15 Choose the correct graph of the function below The scale for each graph is 10 to 10 for the x axis where a tick represents 1 unit and 65 to 65 for the y axis where a tick represents 5 units O A B Q O C y Q 27 5 O D Z Q Q G

let x tan find f s

Answer the following questions for the function x f x b The formula for the first derivative f x can be factored Set f x 0 to find the two critical numbers Hint You can factor out 18 from all terms in the formula for f x The critical values are x Use a comma to separate answers as needed c At the negative critical value listed in part b what is the value of the second derivative Type Integers or simplified fractions Consequently what can be concluded about the graph of f Select the correct choice below and if necessary fill in the answer boxes within your choice OA The graph of f is concave up and f has a relative minimum at OB The graph of f is concave down and f has a relative minimum at OC The graph of f is concave up and f has a relative maximum at OD The graph of f is concave down and f has a relative maximum at OE No conclusion can be made d At the positive critical value listed in part b what is the value of the second derivative Type Integers or simplified fractions Consequently what can be concluded about the graph of f Select the correct choice below and if necessary fill in the answer boxes within your choice OA The graph of f is concave down and f has a relative minimum at OB The graph of f is concave up and f has a relative maximum at 0 OC The graph of f is concave up and f has a relative minimum at OD The graph of f is concave down and f has a relative maximum at E No conclusion can be made e Set the formula for the second derivative f x 0 to find any possible inflection points f x 0 atx What can be concluded about the graph of f at this value of x Select the correct choice below and if necessary fill in the answer boxes within your choice OA There is a point of inflection at where the graph of f changes from concave down to concave up OB There is a point of inflection at where the graph off changes from concave up to concave down OC No conclusion can be made f Choose the graph of the function f x OA Q OB 1500 O C 4400

where b is a real constant Use Desmos to graph this function with a slider for b Guiding Questions 1 What do you notice about how the slope of the function changes with b What does this tell us about how addition or subtraction of constants impact the behavior of functions How does this related to the graph transformations you reviewed in week 1 of the course 2 Now that you have a sense for the function s behavior pick two distinct points a b in the function s domain such that f a f b If the MVT holds what feature must be present in the function for some twith a x b How can we use this observation to get a better

9 To convert from a degrees Celsius to y degrees Fahrenheit we use the formula f x x 32 Find the inverse function if it exists and be sure you could explain its meaning f x Submit answer help formulas

Find the largest domain for f x x 2 2 with upper a bound 6 such that the function f is one to one and non decreasing One to one domain help intervals Your answers must be in interval notation Find the inverse of f restricted to that domain f x help formulas

An object dropped from a height of 500 feet has a height h t in feet after t seconds have elapsed such that h t 500 16t Express t as a function of height h and find the time to reach a height of 100 feet t h time to reach height of 100 feet 4

Let the graph of f x be given below Find the following 5 f 0 f 0 Tiles tmp quick 2023 Click on graph to enlarge help numbers help numbers

Find the inverse of f x 3 3 f x

Let f x f x 2x 7 7x 8 Find f x help formulas

8 The Sum given by a Find b Show geometric the of 5 1 2 1 first term of that three terms the progression terms of certain of the the series is series 4 3 4 3 series Ug are in

Find the interval s of numbers that are less than a distance of 7 away from 8 Enter solution as an interval or union of intervals In absolute value notation where a represents the numbers this is OA z 81 7 OB 2 7 8 help intervals OC x 7 8 OD x 8 7 OE None of the above

A patient needs to keep their blood sugar levels within 10 milligrams per deciliter of 93 milligrams per deciliter What blood sugar levels are okay for that patient Enter solution as an interval or union of intervals Note this is a hypothetical situation help intervals In absolute value notation where xa represents the score on the test this is OA x 10 93 OB x 93 10 OC x 10 93 OD x 93 10 OE None of the above

A machinist must produce a bearing that is within 0 02 inches of the correct diameter of 5 5 inches What are the possible diameters that will work Enter solution as an interval or union of intervals help intervals In absolute value notation where a represents the machined diameter this is OA 5 51 0 02 OB x 0 02 5 5 OC 5 5 0 02 OD x 0 02 5 5 OE None of the above

Find the x intercept s and y intercept s of y 2x 5 2 Enter a point as a b including the parentheses If there is more than one correct answer enter a comma separated list of points x intercept s y intercept s help points help points