Calculus Questions

Determine where the function f x is continuous f x 4 x The function is continuous on the interval Simplify your answer Type your answer in interval notation

An object moves along the y axis marked in feet so that its position at time x in seconds is given by the function f x x 15x 72x a Find the instantaneous velocity function v f x b Find the velocity when x 1 c Find the time s when v 0 a v b The velocity of the object at x 1 is ft s Simplify your answer c The velocity is zero at x seconds Simplify your answer Use a comma to separate answers as needed

Awater sprinkler sends water out in a circular pattern Determine how large an area is watered The radius of watering is 17 ft Use 3 14 for OCEED The area of the circle is Type a whole number or decimal rounded to the nearest hundredth as needed

Find the profit function if cost and revenue are given by C x 130 2 4x and R x 4x 0 01x The profit function is P x

Find each limit Use oo or oo when appropriate 6x 6 x 6 A lim f x x 6 f x 2 B lim f x x 6 OA im f x x 6 A Select the correct choice below and if necessary fill in the answer box to complete your choice OA lim f x Simplify your answer x 6 OB The limit does not exist B Select the correct choice below and if necessary fill in the answer box to complete your choice Simplify your answer C lim f x x 6 BOXER OA lim f x Simplify your answer x 6 OB The limit does not exist s no exist C Select the correct choice below and if necessary fill in the answer box to complete your choice OB The limit does not exist

If 1000 is invested at 7 5 compounded continuously graph the amount in the account as a function of time for a period of 9 years Choose the correct graph A OD AA 2000 1000 0 4 0 2 4 6 8 10 AA 2000 1000 0 02 4 6 8 10 Q Q Q Q 5 OB OE AA 2000 1000 0 02 4 6 8 10 AA 2000 CEED 1000 0 14 02468 10 Q Q Q Q O C OF AA 2000 1000 0 0 2 4 6 8 10 AA 2000 1000 0 0 2 4 6 8 10 Q

Find dy for the function y dy Simplif 650 x

y 2 2 21 Choose the correct graph below OA B 3 5 XXXI do 3 OD

A Calculate the change in Fox from x 8 to x 13 8 Graph F x and use geometric formulas to calculate the area between the graph of F and the x axis from x 8 to x 13 C Verily that your answers from A and B are equal as quaranteed by the fundamental theorem of calculus A Calculate the change in F x from x 8 to x 13 The change is Simplify your answer B Graph F x and use geometric formulas to calculate the area between the graph of F x and the x axis from x 8 to x 13 Choose the graph of F x below OA 400 UND 5000 The area between the graph of F x and the x axis from x 8 to x 13 Simplify your answer 10 C Verity that your answers from A and 8 are equal as guaranteed by the fundamental theorem of calculus Are your answers in part A and part B equal Yes DR O 553

Use geometric formulas to find the unsigned area between the graph of y f x and the x axis over the indicated interval f x x 10 2 11 The area between f x and the x axis is

Give the order of the differential equation below where y represents a function of the variable x x y 2y y 2 Choose the correct answer below 9 OA The given equation is a second order differential equation because it involves a second derivative but no higher derivat OB The given equation is a third order differential equation because it involves a third derivative but no higher derivative OC The given equation is a first order differential equation because it involves a first derivative but no higher derivative OD The given equation is a first order differential equation because it only involves a first derivative

A function f x is given below Express the relationship between f x and f x in words and write a differential equation that f satisfies x 6 CTXH Express the relationship between f x and f x in words Choose the correct answer below OA The derivative of f x is 6 times f x OB The derivative of f x is 1 times e raised to the x 1 power OC The derivative of f x is the same as f x OD The derivative of f x is 1 times f x f x satisfies the differential equation y Type an expression using y as the variable

Isy 19x a solution of the differential equation dx Explain BOCCOD The function y 19x a solution of the differential equation because when it is substituted into the differential equation side of the equation is equal to and the right side of the equation is equal to Simplify your answers

Assuming that x 0 use differentiation to justify the formula dx In x C if Sdx In x C then The derivative of In x is and the derivative of a constant C is so dx In x C for x 0 dx In x C for x 0

Write the following expression in the form ax bx9 where a and b are real numbers and p and q are rational numbers 8 8 x 3 2 8 8 x Simplify your answer Use integers or fractions for any numbers in the expression

3 12 points In the following problem the function f x defined for x 1 is given by the graph below 1 1 2 1 b 3 points Evaluate 2 a 3 points Evaluate T 6f x dx 3 f x 2 dr 4 5 6 7 C 3 points Choose a value of b that would make the definite integral I d 3 points Compute the average value of f x from x 2 to x 6 8 f x dx negative

Graph the equation Then describe the transformation from the parent function f x x 1 y xl G Now describe the transformation from the parent function f x xl Choose the correct answer below OA The parent function f x x is reflected across the x axis OB The parent function f x x is reflected across the y axis OC The parent function f x x is vertically stretched and reflected across the x axis OD The parent function f x xl is vertically compressed and reflected across the x axis G

A model for a company s revenue from selling a software package is R p 2 5p 850p where p is the price in dollars of the software What price will maximize revenue Find the maximum revenue The price maximizes the revenue Simplify your answer The maximum revenue is Simplify your answer

Write an absolute value equation for the graph shown to the right y Simplify your answer 12 10 8

Use the given information to sketch the graph off Domain All real x except x 6 and x 6 9 4 1 0 0 1 9 4 f x 0 on 6 and 6 oo f x 0 on 6 6 f x 0 on 6 and 6 0 f x 0 on 0 6 and 6 co Vertical asymptotes x 6 and x 6 Horizontal asymptote y 0 Choose the correct graph below O A LV OB 12 Q Q 63 O C 1 16 Q Q

Assume f x is continuous on 0 00 Use the given information to sketch the graph off 2 4 7 4 1 0 1 x 1 x X 1 x 4 5 ND 0 0 0 0 ND 2 Choose the correct graph below OA 6 1 OB 0 6 0 5 LICE O C OD

A dairy is planning to introduce and promote a new line of organic ice cream After test marketing the new line in a large city the marketing research department found that the demand in that city is given approximately by the following equation where x thousand quarts were sold per week at a price of Sp each and whose revenue function is given as R x xp p 15 0 x 6 A Find the local extrema for the revenue function 8 On which intervals is the graph of the revenue function concave upward Concave downward A What is are the local maximum a Select the correct choice below and if necessary fill in the answer box to complete your choice OA The local maximum a is are at x Simplify your answer Use a comma to separate answers as needed OB There is no local maximum What is are the local minimumva Select the correct choice below and if necessary fill in the answer box to completeur choice OA The local minimum a is are at x Simplify your answer Use a comma to separate answer OB There is no local minimum swers as needed B On which intervals is the graph of the revenue function concave upward Select the correct choice below and if necessary fill in the answer box to complete your choice OA The graph is concave upward on

The function f x is continuous on 00 00 Use the given information to sketch the graph of f 1 0 20 1 10 0 1 20 20 f x 0 on 0 0 and 20 00 10 0 f x 0 on 10 00 Choose the correct graph off below O A O B f 0 0 f 20 0 f x 0 on 0 20 t x 0 on 0 10 2 O C O D

A candy box is made from a piece of cardboard that measures 13 by 8 inches Squares of equal size will be cut out of each corner The sides will then be folded up to form a rectangular box What size square should be cut from each corner to obtain maximum volume EXXE A square with a side of length inches should be cut away from each corner to obtain the maximum volume Round to the nearest hundredth as needed

Sketch the graph given the following conditions f 3 0 f 9 0 f 2 0 f 3 0 f 9 20 f 2 1 f 3 8 f 9 6

Find all the relative extrema and sketch the grap OF F X x3 6x 10

Find f x f x and f x of f x x 3x 2 8x 4

Suppose that the company sports stylz Inc determines that the cost in dollars from the sale of X sunglasses is given by C X 0 05x 50x Find the average rate of change from selling 300 Phones to and X 301 301 Phones That is X 300 Average rate of change F xa f x X X

T X x 10 x 11 ou

Sketch the graph of the continuous function f that satisfies all of the stated conditions 9 pts brodo hav 201 f 0 2 f 1 0 2 2 f 0 0 f 2 0 bution of for en f x 0 on 0 and 2 00 f x 0 on 0 2 baralimos w f 1 0 6 5 f x 0 on 1 0 f x 0 on 0 1 whoito 120 TOIDUOTOS TOY otrd bouquio pr 18 4 cat 3 A 5 3 2 1 ON 411 2 23 3 4 5 6 2 A 6 305

5 Detailed Clothing s total revenue R x and total cost C x in dollars from the sale of x custom sweatshirts are given as follows 6 pts R x 4x 11x 110 C x 4 2x 5x 10 a Find the profit function P x b Find the number of sweatshirts that must be produced and sold in order to yield the maximum profit c Find the maximum profit

Consider the demand function given by D x 240 20x where D x is the quantity of coffee mugs demanded at a price of x dollars per mug 5 pts a Find the elasticity E x D x D x b At a price of 2 should the company raise or lower the price to increase revenue Explain your answer

11 Find the area between the curves y 4x x and y x 6x 8 Draw the graph and shade the appropriate region 7 pts 10 10 8 6 4 2 98 6 4 2 24 6 2 4 6 8 10

D Suppose an experiment finds that the memorization rate is given by M t 0 009t2 0 2t where M t is the rate in words per minute How many words are memorized in the first 10 minutes 5 pts

the function f x x 3x 1 a Make a sign diagram for the first derivative b Find any relative maximum or relative minimum points Write the points as an ordered pair x y Relative Max Relative Min c Make a sign diagram for the second derivative d Find the inflection point s if there are any Write the points as an ordered pair x Inflection Point s e Graph the function Clearly show any relative maximum relative minimum and inflection points and at least 5 additional points on the graph 10 10 8 6 4 2 thon 8 6 4 2 2 4 6 8 10 2 4 6 8 10 X y toupe no

the following definite and indefinite integrals 14 pts a f x 1 2x 3 dx b 2e 1 dx Tr

3 Find the derivative of the given functions 24 pts a f x 3x 2x 1 Expur 0000 and iw enixolen bno luthsbnow s In 4x

Given the F X axty find the equation of Point 116 tangent line of F X at the Steps M Slope F X lim f xth f x hs0 Find the slope you will get a Find the equation y y m x x the

y 3x 5 and y 2x 3 No Solution Help 3 Submission Data 25 22222 24 23 21 20 19 18 17 16 15 14 13 12 3 11 10 9 8 7 Find the area between them from x 0 to x 3 Clear All Delete Fill WebAssign Graphing Tool

Find the integral by whatever means are necessary either substitution or tables Remember to use absolute values where appropriate Use C for the constant of integration 1 25 x X dx 25 x In 25 1 5 25 x 5 CX

use geometry Sax d dx a 10 b S 4x 1 dx to evaluate the definite integral

The question asks us to sketch the function y cos 3t First note that the function is a cosine function and will have that basic shape From the text the general form of a non shifted cosine function is y a cos bt The text also tells us that a is the amplitude that is the maximum height above zero that the function attains Note that the amplitude must be a positive number Looking at our function the amplitude is 1 1 Step 2 2x Next we need the period From the text the period of the function y a cos bt is b Looking at our function and simplifying the period is of complete cycles the function goes through between 0 and 2 More intuitively the number b gives the number

A patient receives an injection of 9 5 milligrams of a drug and the amount remaining in the bloodstream A t 9 5e 0 04t Find the instantaneous rate of change in milligrams per hour of this amount under the following condition Round your answers to two decimal places a immediately after the injection time t 0 milligrams per hour b after 5 hours milligrams per hour

Tutorial Exercise Sketch the graph of the function showing the amplitude and period of the function below y cos 3t Step 1 The question asks us to sketch the function y cos 3t First note that the function is a cosine function and will ha that basic shape E From the text the general form of a non shifted cosine function is y a cos bt The text also tells us that a is the amplitude that is the maximum height above zero that the function attains Note that the amplitude must be a un number Looking at our function the amplitude is

In a psychology experiment rats were placed in a T maze and the proportion of rats who required more than t seconds to reach the end was as shown below S 0 09e 0 09s ds Use this formula to find the proportion of rats who required more than 9 seconds Round your answer to one decimal place

The number of traffic accidents per year in a city of Population is predicted to be A 0 002 p312 If the population is growing by 500 people per year find the rate at which traffic accidents will be rising when the population is P 40 000

Find where the function is concave up or down include a sign chart and find inflection Points F x x 20 x

suppose that the daily cost in dollars of producing X radius radius at a current production level of 40 racios is C X 0 002x 0 1X 42x 300 C 40 a what is the current daily cost b what is the marginal cust when X 40 C use marginal cost estimate the daily cost of increasing Production to 41 radius daily C 41 C 40 C 40 C 40

Suppose that you meet 20 new people each year but each year you forget 20 of all of the people that you know If y t is the total number of people who you remember after t years then y satisfies the differential equation y 200 2y Do you see why Solve this differential equation subject to the condition y 0 0 you knew no one at birth y

A medical examiner called to the scene of a murder will usually take the temperature of the body A corpse cools at a rate proportional to the difference between its temperature and the temperature of the room If y t is the temperature in degrees Fahrenheit of the body t hours after the murder and if the room temperature is 60 then y satisfies y 0 32 y 60 y 0 98 6 body temperature initially 98 6 a Solve this differential equation and initial condition y b Use your answer to part a to estimate how long ago the murder took place if the temperature of the body when it was discovered was 80 Hint Find the value of t that makes your solution equal 80 Round your answer to two decimal places t hr