Calculus Questions

Use the exponential decay model A Aekt to solve the following The half life of a certain substance is 20 years How long will it take for a sample of this substance to decay to 66 of its original amount It will take approximately Round to one decimal place years percent for the sample of the substance to decay to 66 of its original amount percent per year

Write out the first five terms of the sequence with 1 n 1 converges and if so find its limit Enter the following information for an a1 a2 a3 a4 ab In n lim n on 1 Enter DNE if limit Does Not Exist Does the sequence converge ln n n 1 Enter yes or no n 1 determine whether the sequence

Find all critical points of the following function f x y 3 X 3 3 y 3 9xy What are the critical points Select the correct choice below and fill in any answer boxes within your choi OA The critical point s is are Type an ordered pair Use a comma to separate answers as needed OB There are no critical points

Some questions make use of the Interactive Graphing Tool to draw a basic graph and then use interactions to edit special graph values in order to complete the graph Click the icon to view instructions for the Interactive Graphing Tool 2 Use the x cubed tool to graph the x cubed function Update the function to have a Vertical Stretch of 3 a Horizontal Stretch of 1 a Vertical Shift of 0 a Horizontal Shift of 5 and a Reflection over the x axis Use the x cubed tool button on the graphing palette to place the function on the graph Then update the Vertical Stretch Horizontal Stretch Vertical Shift Horizontal Shift and Reflect over x axis interactions to have the values noted above Click to enlarge

4 Al 3 the function given by f x z if x 3 4 2x 1 if x 3 2 is A undefined B continuous but not differentiable S C differentiable but not continuous D neither continuous nor differentiable E both continuous and differentiable lim f x flam Jim 4 78

Use a graphing utility to approximate the solutions of the equation in the interval 0 2 Round your answers to three decimal places Enter your answers as a comma separated list 5 sin x 3 cos x 0

Use a graphing utility to approximate the solutions to three decimal places of the equation in the given interval Enter your answers as a comma separated list 6 sec x tan x 9 0 1 2 1 2 X SPEHE

27 Find the exact area under y 5te for 0 t 6 NOTE Enter the exact answer Area

A university is trying to determine what price to charge for tickets to football games At a price of 16 per ticket attendance averages 40 000 people per game Every decrease of 2 adds 10 000 peopl the average number Every person at the game spends an average of 4 00 on concessions What price per ticket should be charged in order to maximize revenue How many people will attend at that price

The parametric equations and parameter intervals for the motion of a particle in the xy plane are given below Identify the particle s path by finding a Cartesian equation for it Graph the Cartesian equation Indicate the portion of the graph traced by the particle and the direction of motion x 5 cosht y 5 sinh t costs Find a Cartesian equation for the particle s path y x 25 Type an exact answer using radicals as needed Graph the Cartesian equation below Indicate the direction of motion as t increases Use a dotted line to represent portions if any of the graph that are not traced by the particle Choose the correct graph below O A 10 10 10 OB 10 10 10 10 Q C 10 10 10 10 Q O D 10 10 10

The equation below gives parametric equations and parameter intervals for the motion of a particle in the xy plane Identify the particle s path by finding a Cartesian equation for it Graph the Cartesian equation Indicate the portion of the graph traced by the particle and the direction of motion x 4t 3 y 16t t Graph the Cartesian equation below Indicate the direction of motion as t increases Choose the correct graph below A Q B 40 C OA The portion of the graph where t is less than B None of the graph C The entire graph O D The portion of the graph where t is greater than Y G Indicate the portion of the graph traced by the particle Select the correct choice below and if necessary fill in the answer box to complete your choice

Now we substitute our function of x back into our equation for u and w In 9x 1 1 C x Step 6 Now we know the fixed costs are 250 Thus we have the following C 0 250 Plugging this into the cost function we can find K 250 C 0 n 9 250 1 K 250 1 K In 9x 1 K 2251 X 1 K

A company s marginal cost function is MC x The integral we need is thus C x Step 1 We want to find the cost function from the marginal cost From the text we know that the cost is the integral of the marginal cost Solving for dx we have dx 1 9 Step 3 Setting u 9x 1 we find the differential du 9x 1 C x dx C u MC x dx Step 2 Since the integrand is somewhat complicated we try a u substitution The denominator is a good guess so we try that first o and its fixed costs are 250 Find the cost function 1 1 9 du Substituting u and du into our integral we have the following 9x 1 9x 1 X du 1 9x 1 dx dx

Find parametric equations and a parameter interval for the motion of a particle that starts at a 0 and traces the circle x y a a once clockwise c three times clockwise c Which of the following sets of equations could trace the circle x y a three times clockwise starting at a 0 O A x asin t y a cos t 0 t 4 B X a cos t ya sin t 0 t 8 O c x a cos t y a sin t 0 t 2 D x a cos t yasin t 0 t 6 b once counterclockwise d three times counterclockwise d Which of the following sets of equations could trace the circle x y a three times counterclockwise starting at a 0 O A x asin t y a cos t 0 t 4 OB X a cos t y a sin t 0 t 2n O c x a cos t y a sin t 0 t 8 D X a cos t y asin t 0 t 6

Find the fixed point of the function f A fixed point of a function f is a real number c such that f c c Round your answer to three decimal places f x cos 2x CM radians

Approximate the x intercepts of the graph Use a graphing utility to check your solutions Enter your answers as a comma separated list Let be any y 4 sin xx 4 cos x X X

Solve the multiple angle equation Enter your answers as a csc 2 2 X Need Help Submit Answer 9 1 Points DETAILS Approximate the x intercepts of the graph Use a graphing utility to check your solutions Enter your answers as a co y 4 sin x 4 cos x X Read It Need Help Submit Answer 1 5 Read It LARPRECALCRMRP7 5 3 070 Watch It 1 2 3

Use a graphing utility to approximate the solutions of the equation in the interval 0 2x by collecting all terms on one side graphing the new equation and using the zero or root feature to approximate the x intercepts the graph Round your answers to four decimal places Enter your answers as a comma separated list If there is no solution enter NO SOLUTION 2 sec x tan x 3 X

A company s marginal cost function is MC x given below where x is the number o Find the total cost of the first hundred units 6e 0 022 MC x

5 Find the equa 6 To find the derivative of h x x 2e 6 would require the chain rule What functions f x and g x could you use to find h x if h x f g x

Solve the equation for x cot r 1 0 STEP 1 Subtract the constant from both sides cot x 1 0 cot x STEP 2 The term cot x is now isolated in the equation Find a particular solution on the interval 0 0 x X STEP 3 On 00 00 there are an infinite number of solutions Write the general form of the solution Use n as an integer con X

5 Find the equation of the tangent line to f x 10x 6 when x 1 6 ohoin rule What functions f x and g x

Verify that each x value is a solution of the equation 3 csc x 12 csc x 0 a X TU 6 3 csc 7 12 csc b x 5r 6 sc 7 6 0 3 csc 5 12 csc 5 6 6 R 48 48

Solve the multiple angle equation Enter your answers as a comma separated list Let n be any integer sin 7x X 3 2

Find all solutions of the equation in the interval 0 360 Enter your answers from smallest to largest tan x 3 O

Graphing Directions Answer all three parts Consider the function f x graphed to the right f x 2 x 3 4 Part A Record all of the transformations of f x from the parent function g x x 10 Domain 0 Range 4 2 2 4 0 10 46 Partner B Determine the domain and range of the function Record your answer using set notation 10 12 14 16 18 Partner C For what interval s on the domain is the function increasing or decreasing How d you know Record your answer using interval notation What interval is positive What interval is negative How do you know

Directions Be sure to answer all A function is given by x 1 Positive f x Part A Find the average rate of change on the interval 1 8 AL 10 Part B For what interval s on the domain is the function increasing or decreasing Record your answer using interval notation Part C For what interval s on the domain is the function positive and negative Record your answer using interval notation

Piecewise Defined Functions Graph the functions in Exercises 25 28 Jx 25 f x 26 g x 27 F x 28 G x 2 x f1 x 12 X 0 x 1 1 x 2 4 x 33 2x 2x 53 93 1 x x 0 x 1 1 x 2 x 1 x 0 0 x

Find the slope of the tangent line to the ellipse slope 1 at the point x y Are there any points where the slope is not defined Enter them as semicolon separated ordered pairs e g 1 3 2 5 Enter none if there are no such p slope is undefined at

Find dy da in terms of x and y if cos 7y sin 7y y 14 0 da

Co US Secti Consider the equation below If an answer does not exist enter DNE f x x4 8x 7 a Find the interval on which f is increasing Enter your answer using interval notation 2 Find the interval on which f is decreasing Enter your answer using interval notation b Find the local minimum and maximum values of f local minimum value 9 local maximum value 7 Par c Find the inflection points 2 V 3 2 x y 3 3 x smaller x value BB x larger x value X Find the interval on which fis concave up Enter your answer using interval notation 2 2 U 00 3 Find the interval on which fis concave down Enter your answer using interval notation 2

If 4x 4x xy 3 and y 3 15 find y 3 by implicit differentiation

41 46 Evaluate the integral by interpreting it in terms of areas 42 f 2x 1 dx 44 2x 1 dx 46 2x 16 x dx 41 10 5x dx 2 43 4 15 1 9 x dx 3 x dx

35 The graph of f is shown Evaluate each integral by inter preting it in terms of areas a f f x dx c ff x dx e ff x dx YA 2 0 2 b ff x dx d ff x dx f f f x dx y f x 4 6 8 AX x

27 34 Use the form of the definition of the integral given in Theorem 4 to evaluate the integral 28 f x dx 27 f 3x dx

19 22 Express the limit as a definite integral on the given interval exi n11 Xi 19 lim n Ax 0 1 20 lim x 1 1x Ax 2 5

radius loses 3 inches every 30 seconds Do the following a Express the circumference in inches of the circle as a function of its radius in inches way that C r b Express the radius in inches of the circle as a function of time in minutes c Express the circumference in inches of the circle as a function of time in minutes C t d Choose the composition that represents your answer to part c O C t Cor t O r t ro C t O C r ro C r O C t ro C t Or C ro C t

Use the graph of y g x below to determine which of the choices could be the graph of y g x Hint Make a sign chart and interpret it first

Use the graph of y f x below to determine which of the choices could be the graph of y f x Hint Make a sign chart and interpret it first

A waste management company is designing a rectangular construction dumpster that will be twice as long as it is wide and must hold 20 yd of debris Find the dimensions of the dumpster that will minimize its surface area X

Drum Tight Containers is designing an open top square based rectangular box that will have a volume of 62 5 in What dimensions will minimize surface area What is the minimum surface area

ROO OC

Graph the follwing function by hand and support your sketch with a calculator graph Give the domain range and the equation of the asymptote Determine if f is increasing or decreasing on its dom 1 x 10 sur Give the domain range and equation of the asymptote The domain is 00 00 Type your answer in interval notation The range is 0 00 Type your answer in interval notation The equation of the asymptote is y 0 Type an equation Determine if f is increasing or decreasing on its domain Choose the correct answer below OA The function is increasing on its domain because as the value of x increases the y value remains the same B The function is increasing on its domain because as the value of y increases the y value decreases OC The function is increasing on its domain because as the value of x increases the y value increases D The function t is decreasing on its domain because as the value of x increases the y value decreases Q Q

An open top bin is to be made from a 15 centimeter by 40 centimeter piece of plastic by removing a square from each corner of the plastic and folding up the flaps on each side What size square should be cut out of each corner to get a bin with the maximum volume Enter an exact answer for the area of the square

estimate the total amount of infection needed to develop symptoms of measles NA N f t 1000 ZA 0 12 21 t days

A car rental company charges its customers p dollars per day to rent a car where 35 SPS 120 The number of cars rented per day can be modeled by the linear function p 1200 10p How much should the company charge each customer per day to maximize revenue Do not include units or a dollar sign in your answer

x X 4 OA 8 OB Oc

The equation below gives parametric equations and parameter intervals for the motion of a particle in the xy plane Identify the particle s path by finding a Cartesian equation for it Graph the Cartesian equation Indicate the portion the graph traced by the particle and the direction of motion x 6t 1 y 24t 2 t y 4x 2 Graph the Cartesian equation below Indicate the direction of motion as t increases Choose the correct graph below O A OB O C 10 10 Indicate the portion of the graph traced by the particle Select the correct choice below and if necessary fill in the answer box to complete your choice OA The entire graph B The portion of the graph where t is less than D C None of the graph D The portion of the graph where t is greater than

For y In 7 3x 9 to find y would require the chain rule If y f g x find an f x and g x that would allow you to use the chain rule

A bike rental company charges its customers p dollars per day to rent a bike where 5 p 30 The number of bikes rented per day can be modeled by the linear function n p 300 10p How much should the company charge each customer per day to maximize revenue Do not include units or a dollar sign in your answer