Calculus Questions

An inverted cone has a height of 11 inches and an initial radius of 18 inches The volume of the inverted cone is decreasing at a rate of 541 cubic inches per second with the height being held constant What is the rate of change of the radius in inches per second when the radius is 5 inches Submit an exact answer Do not forget to include a negative sign if the radius is decreasing Remember that the volume of a cone is V r h Provide your answer below

Find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit Assume that revenue R x and cost C x of producing x units are in dollam R x 60x 0 1x C x 4x 15 In order to yield the maximum profit of units must be produced and sold Simplify your answers Round to the nearest cent as needed COLD

Of all numbers that add to 82 find the pair that has the maximum product That is maximize the objective function Q xy subject to the constraint x y 82 The values of x and y that have the maximum product are x Type integers or simplified fractions The maximum product of x and y is Q Type an integer or a simplified fraction and y B

Find the absolute maximum and minimum values of each function over the indicated interval and indicate the x values at which they occur f x 3x4 2x 2 2 The absolute maximum value is at x Use a comma to separate answers as needed The absolute minimum value is at x 0 Use a comma to separate answers as needed

An employee s monthly productivity M in number of units produced is found to be a function of the number t of years of service For a certain product a productivity function is shown below Find the maximum productivity and the year in which it is achieved M t 3t 144t 140 0 t 40 The maximum productivity is achieved in year The maximum productivity is units

Question A circle has a radius that is increasing at a rate of JT second What is the rate of change of the area of the circle when the radius is 3 feet Do not include the units in your answer feet per

What is lim p x 818 2x P x 1 4

Graph the tangent line to the function defined implicitly by the equation below at the point 1 2 3x x 2 2y y The black dot has already been placed at the desired point on the curve The green dot should be placed at another point along the tangent line Provide your answer below 10 14 10 0 10 10 5 2

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5 km Coral reef 15 km 1 Provide your answer below Tent Shoreline A visitor is staying in a tent that is 15 kilometers east of the closest point on a shoreline to a coral reef The coral reef is 5 kilometers due south of the shoreline The visitor plans to travel from the tent to the coral reef by running and swimming If the visitor runs at a rate of 9 kmph and swims at a rate of 5 kmpl how far should the visitor run to minimize the time it takes to reach the coral reef Round to two decimal places

Find the slopes of the curve y4 y x at the two points shown here 4 1 L 3 4 12 X HE The slope of the curve y y x at the point 3 is 4 2 Type an exact answer using radicals as needed The slope of the curve 3 3 4 2 ve y4 y x at the point 3 is Type an exact answer as needed sing radicals

dy Write the function in the form y f u and u g x Then find as a function of x dx y 8x U Type an expression using x as the variable y Type an expression using u as the variable dy dx Type an expression using x as the variable

Determine algebraically whether the given function is even odd or neither g x 8x 6 O Even ONeither O Odd

Write an equation of a circle with diameter AB A 5 5 B 15 15 Choose the correct answer below OA x 10 y 10 15 OC x 5 y 5 15 O E x 10 y 10 5 OG x 10 y 10 50 OB x 10 y 10 50 OD x 10 y 10 50 OF x 15 y 15 5 OH x 10 y 10 50

Find the value of x X Do not include the degree symbol in your answer Round to the nearest degree as need

Find the area of the shaded region Use 3 14 16 m r 8 m 16 m m Round the final answer to the nearest hundredth as needed Round all intermediate values to the nearest hundre as needed

Question A Use implicit differentiation to find the equation of the tangent line to the function defined implicitly by the equation below at the point 1 1 3x 4x 10 2y y Give your answer in the form y mx b

Find an equation for the line tangent to the curve at the point defined by the given value of t Also find the value of d y 2 dx at this point x 16 sint y 4 cos t t The equation TU represents the line tangent to the curve at t Type an exact answer using radicals as needed T 4

A car rental company charges its customers p dollars per day to rent a car where 35 p 175 The number of cars rented per day can be modeled by the linear function n p 700 4p How much should the company charge each customer per day to maximize revenue Do not include units or a dollar sign in your answer Provide your answer below

Find the area under one arch of the cycloid x 2a t sin t y 6a 1 cos t The area is Type an exact answer using as needed

Rock formation 5 mi Select the correct answer below 2 89 miles 4 11 miles 7 mi A rock formation is 5 miles due north of its closest point along a straight shoreline A visitor is staying in an inn that is 7 miles east of that point The visitor is planning to go from the inn to the rock formation Suppose the visitor runs at a rate of 6 mph and swims at a rate of 3 mph How far should the visitor run to minimize the time it takes to reach the rock formation 5 56 miles 5 77 miles Inn Shoreline

Find the length of the curve JU x 5 cost 5t sint y 5 sin t 5t cost 0 t 2 The length is units Type an exact answer using as needed

Find the length of the following curve 2t 11 2 3 X y 5t The length of the curve is 2 0 t 4 Simplify your answer

Find the length of the curve 2t 1 3 2 3 X 2 2 y 0 t 12 The length of the curve is Simplify your answer

Find the length of the curve x t y 1 2 0sts 3 The length of the curve x t y 2 on 0sts 3 is is 0 Type an integer or a fraction

Find an equation for the line tangent to the curve at the point defined by the given value of t Also find the value o d y 2 dx at this point x 41 10 y t t 1 Write the equation of the tangent line

Using a rock wall as one side and fencing for the other three sides a rectangular patio will be constructed Given that there are 172 feet of fencing available determine the dimensions that would create the patio of maximum area and identify the maximum area Enter only the maximum area Do not include units in your answer

Question Find the equation of the tangent line at the point 1 2 to the curve defined implicitly below x 3x 6 2y 3y

Question Find the equation of the tangent line at the point 1 2 to the curve defined implicitly below x x y 3

Question If you want to produce an open top box from a 16 in by 22 in flat piece of cardboard what is the maximum possible volume in in of the box Round your answer to the nearest whole number and omit units Provide your answer below

4 Carry out a simulation experiment using a statistical computer package or other software to study the sam pling distribution of when the population distribution is Weibull with a 2 and 3 5 as in Example 5 20 Consider the four sample sizes n 5 10 20 and 30 and in each case use 1000 replications For which of these sample sizes does the X sampling distribution appear to be approximately normal

Question Determine lim 2x X18

nd the quotient Leave the result in trigonometric form Let 0 0 2 9 cos 4 4 i sin 4 4 7 cos 1 5 i sin 1 5

Find all absolute maximum and minimum values for the function y 3x 6x 9 on the interval 5 1 enter NONE if there are none help numbers enter NONE if there are none help numbers a Absolute maximum is b Absolute minimum is

Find the global maximum value of f x 6ce 2 on the interval 6 5 The global maximum occurs at x 0 and the maximum value is y

Absolute Extreme Values Find the absolute maximum and minimum values of the function f x e on the interval 4 7 Absolute maximum value Absolute minimum value

GIA STATE The administration conducted a survey to determine the proportion of students wh ride a bike to campus Of the 128 students surveyed 6 ride a bike to campus Which of the following is a reason the administration should not calculate a confidence interval to estimate the proportion of all students who ride a bike to campus Check all that apply The sample needs to be random but we don t know if it is The actual count of bike riders is too small The actual count of those who do not ride a bike to campus is too small

6 Let be a function that is differentiable on the open interval 1 10 If f 2 5 f 5 5 and f 9 5 which of the following must be true I f has at least 2 zeros II The graph of f has at least one horizontal tangent III For some c 2 e 5 f c S A None B I only C I II only D I III only E I II III

K shifts Midpoints and quarter points are identified by dots The simplest form of an equation for the graph is y Use integers or fractions for any numbers in the expression 3x 12 3x 4 2 0 12 3t

ell T 1 2 Statement ZU ZV TW LUV ZTWU ZTWV TW TW SATUW ATVW W Complete the proof that ATUWATVW V Reason Given Given Additive Property of Angle Measure Additive Property of Angle Measure Additive Property of Length All right angles are congruent Angles forming a linear pair sum to 180 Definition of angle bisector Definition of equilateral triangle

2 12 pts Solve the inequality Show your work sign analysis 5 7 X 3 X 4

Find the global maximum and global minimum values of the function g 0 60 7 sin 0 on the interval 0 Use the derivatives to algebraically answer the question and leave answer in 4 decimal places The global minimum value I 0 5410 The global maximum value 6 Question Help Video X

3 Dudley is dropped out of a plane and falls vertically toward the earth Dudley s acceleration is the sum of two components a downward acceleration due to gravity equal to g 9 8 kg m s an acceleration in the opposite direction to Dudley s current velocity and proportional to this velocity a Write a differential equation that describes Dudley s vertical velocity as a function of time Choose downward as the positive direction There will be an as yet undetermined constant of proportionality involved call it k b Compute an explicit formula for Dudley s velocity as a function of time assuming that i Dudley will approach a maximum vertical speed of 65 m s and ii that Dudley is shoved out of the plane with an initial downward velocity of 5 m s

Consider the function f x 4 4x question The absolute maximum value is and this occurs at x The absolute minimum value is and this occurs at x 4 x 1 Use the derivatives to algebraically answer the

Consider the function f x 6x 4x 4 0 x 10 Use the derivatives to algebrically answer the question The absolute maximum of f x on the given interval is at x and the absolute minimum of f x on the given interval is at

Consider the function f x x 72x 2 5 x 13 Use the derivatives to algebrically answer the question This function has an absolute minimum value equal to and an absolute maximum value equal to

Let f 0 sin 0 3 cos 0 Calculate the derivative Note to get the to show up type theta and the system will immediately convert the word to the symbol f 0 cos 0 3 sin 0 List the first interval after 0 where both endpoints of the interval are critical numbers of the function on which f is increasing Increasing List the first interval after 0 where both endpoints of the interval are critical numbers of the function on which f is decreasing Decreasing

Consider the sequence Describe the behavior of the sequence Is the sequence monotone Monotone decreasing for all n Is the sequence bounded an 2 5 0 9 Bounded Determine whether the sequence converges or diverges If it converges find the value it converges to diverges enter DIV

Determine whether the sequence is divergent or convergent If it is convergent evaluate its limit If it diverges to infinity state your answer as INF without the quotation marks If it diverges to negative infinity state your answer as MINF If it diverges without being infinity or negative infinity state your answer as DIV lim 7n 11 700 11 12

Determine whether the sequence is divergent or convergent If it is convergent evaluate its limit If it diverges to infinity state your answer as inf If it diverges to negative infinity state your answer as inf If it diverges without being infinity or negative infinity state your answer as div lim 1 sin 17 n 11 700