MTH241 1.7-2.1 Quiz

An analysis of the daily output of a factory assembly line shows that about 40t+t^2−148t^3 units are produced after t hours of​ work, 0≤t≤8. What is the rate of production​ (in units per​ hour) when t=4​?

At t=4​, the rate of production is _____ units per hour. -Find the derivative then plug in 4

Find the first derivative. f(t)=(t^5+3)^8

Differentiate using the chain rule. Find derivative of (t^5+3)^8 and the derivative of (t^5+3)

Suppose the revenue from producing​ (and selling) x units of a product is given by R(x)=9x−.03x^2 dollars

Find the marginal revenue at a production level of 30. -Find derivative of equation, 9-0.06x and input 30, 7.2 Find the production levels where the revenue is ​$600. -Set equation equal to 600, then get equation to equal 0 and use quadratic formula. You will get two answers for this question.

Let f(t)=3t+1/4t. Find f′′(−2).

First find f'(t), which is 3-1/4t^2. Next find f"(t), which is 1/2t^3 Finally, find f"(-2), which is 1/2(-2)^3 or -1/16

Compute the following. d/dz: (z^2+3z+2)^7 at z=−1

First find the derivative: d/dz(z^2+3z+2)^7 x d/dx(z^2+3z+2) which is 7(z^2+3z+2)^6(2z+3) Input -1 for z, which is 0

Let f(t) be the temperature​ (in degrees​ Celsius) of a liquid at time t​ (in hours). The rate of temperature change at time a has the value f′(a). Determine the proper method of solution for the question. How fast is the temperature of the liquid changing after 2 hours​?

For which of the following is the question​ asking? -The instantaneous rate of change at a certain time To find this at a certain​ time, which of the following must be​ evaluated? -f′(a) at a certain time a What is the value of t that is needed to answer the​ question? -2 Therefore, which method of solution is used to determine how fast the temperature of the liquid is changing after 2 ​hours? -Compute f′(2).

Suppose a company finds that the revenue R generated by spending x dollars on advertising is given by R=1100+87x−.03x2​, for 0≤x≤1500. Find dR/dx x=1200.

Just find derivative and input x

At the end of the holiday season in​ January, the sales at a department store are expected to fall. It was initially estimated that for the x day of​ January, the sales will be​ S(x). The financial analysts at the store corrected their projection and are now expecting the total sales for the x day of January to be​ T(x). S(x)=5+25/(x+1^)2 thousand dollars T(x)=32/3+28/3(3x+1)^2 thousand dollars Compute T(1)​, T′(1)​, S(1)​, and S′(1) and interpret the results.

T(1)=32/3+28/3(3(1)+1)^2=11.25 T'(1)=

Let​ C(x) be the cost​ (in dollars) of manufacturing x bicycles per day in a certain factory. If C(50​)=6000 and C′(50)=75​, estimate the cost of manufacturing 51 bicycles per day.

The cost of manufacturing 51 bicycles per day would be approximately $6075. Add cost of 6000 plus 75

Suppose that f(x)=5x^2. ​(b)What is the​ (instantaneous) rate of change of​ f(x) when x=2​?

To compute the​ (instantaneous) rate of change of f(x) when x=2​, first compute f'(x). f'(x)=10x, then 10(2)=20.

Find d/da: x^2a^3+y^5a+z^4.

Treat everything but a as a constant

Suppose that f(700)=4000 and f′(700)=20. Estimate each of the following. ​(a) f(701) ​(b) f(700.5) ​(c) f(699) ​(d) f(698) ​(e) f(699.75)

Use equation f(x)=f(a)+f'(a)(x-a) For f(701)=f(700)+f'(700)(701-700) f(701)=4000+20(1)=4020

The third derivative of a function​ f(x) is the derivative of the second derivative f′′(x) and is denoted by f′′′(x). Compute f′′′(x) for the following function. ​f(x)=4x^5-3x^4+6x

f'(x)=20x^4-12x^3+6 f''(x)=80x^3-36x^2 f'''(x)=240x^2-72x

Suppose that s=Tx^3+6xG+T^3. Find ​(a) ds/dx ​(b) ds/dG ​(c) ds/dT

(a) Although the expression Tx3+6xG+T3 contains several​ letters, the notation ds/dx indicates that for purposes of calculating the derivative with respect to​ x, all letters except x are to be considered as constants. Same for rest

A helicopter is rising straight up in the air. Its distance from the ground t seconds after takeoff is s(t) ​feet, where s(t)=3t^2+6t. ​(a) How long will it take for the helicopter to rise to 24 ​feet? ​(b)Find the velocity and acceleration of the helicopter when it is 24 feet above the ground.

(a) The time it will take for the helicopter to rise to 24 feet is _____ seconds. -Set the equation equal to 24 and solve. Use a positive number. (b) When the helicopter is 24 feet above the​ ground, its velocity is ____ feet per second. -Find derivative and plugin 2. 6t+6, 6(2)+6=18 (c) When the helicopter is 24 feet above the​ ground, its acceleration is ____ feet per second squared. -Find second derivative, 6

An object moving in a straight line travels s(t) kilometers in t​ hours, where s(t)=8t^2+3t. ​(a) What is the​ object's velocity when t=7​? ​(b) How far has the object traveled in 7 ​hours? ​(c) When is the object traveling at a rate of 7 ​km/h?

(a) The​ object's velocity when t=7 is ____ km/h. -Velocity is derivative of the equation, input 7. 16(7)+3=115 (b) The distance the object traveled in 7 hours is ____ km. -Find S(7), 8(7)^2+3(7)=413 ​(c) The time at which the object is traveling at a rate of 7 ​km/h is t=_____ hour. -Set derivative equal to 7, 16t+3=7, t=1/4

If f(x)=4x^2+3x​, calculate the average rate of change of​ f(x) over the following intervals. ​(a) 4≤x≤5 ​(b) 4≤x≤4.5 ​(c) 4≤x≤4.1

(a) Which expression can be used to find the average rate of change of f(x) over the interval 4≤x≤5​? -f(5)−f(4)/5−4 The average rate of change of f(x) over the interval 4≤x≤5 is ____. (115-76)/(5-4)=39/1=39 Which expression can be used to find the average rate of change of f(x) over the interval 4≤x≤4.5​? f(4.5)−f(4)/4.5−4 The average rate of change of f(x) over the interval 4≤x≤4.5 is ____. (94.5-76)/(4.5-4)=37

Let f(t) be the temperature​ (in degrees​ Celsius) of a liquid at time t​ (in hours). The rate of temperature change at time a has the value f′(a). Determine the proper method of solution for the question. What is the temperature of the liquid after 21 hours​ (that is, when t=21​)?

For which of the following is the question​ asking? A. The solution to f(t)=0 B. f′′(c) at a certain time c C. f′(a) at a certain time a D. f(t) at a certain time t What is the value of t that should be substituted into the​ function? 21 Therefore, which method of solution is used to determine the temperature of the liquid after 12 hours? Compute f(21)

A particle is moving in a straight line in such a way that its position at time t​ (in seconds) is s(t)=t^2+2t+4 feet to the right of a reference​ point, for t≥0. ​(a) What is the velocity of the object when the time is 5 ​seconds? ​(b) Is the object moving toward the reference point when t=5​? Explain your answer. ​(c) What is the​ object's velocity when the object is 12 feet from the reference​ point?

The velocity function to be used to calculate the​ object's velocity is v(t)=_____. 2t+2 (a) The​ object's velocity when t=5 is 12 feet per second (2(5)+2)=12 (b) The object is not moving towards the reference point when t=5 because the velocity at t=5​, 12 feet per second is positive, which indicates the object is moving away from the reference point. (c) The​ object's velocity when the object is 8 feet from the reference point is _____ -Set the original equation equal to 8 and factor out, use positive number and plugin to derivative eqaution.

Let​ C(x) be the cost​ (in dollars) of manufacturing x bicycles per day in a certain factory. Interpret C(60​)=7000 and C′(60)=75.

What does C(60​)=7000 represent in the context of this​ problem? -The cost of producing 60 bicycles is $7000. What does C′(60)=75 represent in the context of this​ problem? -The approximate cost of producing the `61st bicycle is $75.

Let​ f(t) be the temperature of a cup of coffee t minutes after it has been poured. Interpret f(7​)=110 and f′(7)=−6. Estimate the temperature of the coffee after 7 minutes and 30 seconds, that​ is, after 7.5 minutes.

What does ​f(7​)=110 ​imply? -7 minutes after the coffee has been​ poured, the temperature of the cup of coffee is 110 degrees. What does f′(7)=−6 ​imply? -7 minutes after the coffee has been​poured, the temperature of the cup of coffee is falling at a rate of 6 degrees per minute. After 7 minutes and 30 seconds, the coffee will be _____ degrees. 107 degrees (30 seconds is half a minute so half of -6 is -3)

Suppose that 5 mg of a drug is injected into the bloodstream. Let f(t) be the amount present in the bloodstream after t hours. Interpret f(1)=4 and f′(1)=−0.6. Estimate the number of milligrams of the drug in the bloodstream after 1 1/4 hours.

What is the meaning of f(1)=4​? -1hour after the drug was​ injected, the amount present in the bloodstream is 4 mg. What is the meaning of f′(1)=−0.6​? -1 hour after the drug was​injected, the amount present in the bloodstream is falling at a rate of 0.6 mg per hour After 1 1/4 ​hours, the number of milligrams of the drug in the bloodstream will be ____ mg. 4-0.6(1.25-1)=3.85

Let f(t) be the temperature​ (in degrees​ Celsius) of a liquid at time t​ (in hours). The rate of temperature change at time a has the value f′(a). Determine the proper method of solution for the question. By how many degrees did the temperature rise during the first 10 hours​?

Which of the following will result in the number of degrees the temperature of the liquid rose during the first 10 hours​? -Subtract the​ liquid's initial temperature from its temperature 18 hours later. What is the​ liquid's initial​ temperature? -f(0) What is the​ liquid's temperature 18 hours​ later? -f(18) ​Therefore, which method of solution is used to determine the number of degrees the temperature of the liquid rose during the first 18 hours​? -Compute f(18)−f(0).

At the end of the holiday season in​ January, the sales at a department store are expected to fall. It is estimated that for the x day of January the sales will be ​S(x)=3+9/(x+1)^2 thousand dollars. ​a) Find the total sales for January 12 and determine the rate at which sales are falling on that day. ​b) Compare the rate of change of sales on January 2 to the rate on January 12. What can you infer about the rate of change of​ sales?

a) The total sales for January 12 were ___ thousand dollars Input 12 into the S(x) equation to get 3.053 The sales are falling at a rate of ____ thousand dollars per day on January 12. Find the derivative of S(x) and input 12, to get -0.008 ​b) Which of the following statements are correct about the rate of change of sales on January 2 and January 12​? Look at the slopes of the graph.

A toy rocket fired straight up into the air has height ​s(t)=160t−16t^2 feet after t seconds. a. What is the​ rocket's initial velocity​ (when t=​0)? b. What is the velocity after 4 ​seconds? c. What is the acceleration when t=9​? d. At what time will the rocket hit the​ ground? e. At what velocity will the rocket be traveling just as it smashes into the​ ground?

a. The​ rocket's initial velocity is _____ ft/sec. -Find derivative and plugin 0, equal 160 b. After 4 seconds, the velocity of the rocket is _____ ft/sec. -Find derivative and plugin 4, equals 32 c. When t=9​, the acceleration of the rocket is ____ ft/sec2. -Find the second derivative, -32 d. The rocket will hit the ground ____ seconds after it is launched. -Set the original equation equal to 0, factor, and use a larger number. 16t(10-t)=0, t=10 e. When the rocket smashes into the​ ground, it will be travelling at _____ ft/sec. -Plug d answer into the velocity (derivative) equation. 160-32(10)=-160

Let​ R(x) denote the revenue​ (in thousands of​ dollars) generated from the production of x units of computer chips per​ day, where each unit consists of 100 chips. ​(a) Represent the following statement by equations involving R or R′​: ​"When 1300 chips are produced per​ day, the revenue is ​$26,000 and the marginal revenue is $.75 per​ chip." ​(b) If the marginal cost of producing 1300 chips is ​$1.5 per​ chip, what is the marginal profit at this production​ level?

​(a) Choose correct equations below. R(13)=26​, R′(13)=.075 (b) When 1300 chips are​ produced, the marginal profit is ____dollars per chip. Marginal Profit (P')=Revenue (R')-Cost (C') P'(x)= R'(13)-C'(13) P'(x)=0.75-1.5= -1.425

Let​ S(x) represent the total sales​ (in thousands of​ dollars) for the month x in the year 2005 at a certain department store. Represent each statement to the right by an equation involving S or S′.

​(a) The sales at the end of January reached $115,430 and were rising at the rate of $1300 per month. S(1)=115.430, S′(1)=1.3 ​(b) At the end of October​, the sales for this month dropped to $80,000 and were falling by about $400 a day.​ (Use 1 month=30 days.) 400 x 30=12000 so, S(10)=80 and S'(10)=-12

​(a) Let​ A(x) denote the number​ (in hundreds) of computers sold when x thousand dollars is spent on advertising. Represent the following statement by equations involving A or A′​: When three thousand dollars were spent on advertising the number of computers sold was 1400 and it was rising at the rate of 70 computers for each 1000 dollars spent on advertising. ​(b) Estimate the number of computers that will be sold if ​$4000 is spent on advertising.

​(a) Which equations best represent the​ statement? A(3)=14 and A′(3)=0.7 ​(b) The number of computers that will be sold if ​$4000 is spent on advertising is approximately ___. -Compute A(4), A(4)=A(3)+A'(3), 14+0.7=14.7 or 1470

Consider the cost function ​C(x)=6x^2+16x+24 ​(thousand dollars). ​a) What is the marginal cost at production level x=3​? ​b) Use the marginal cost at x=3 to estimate the cost of producing 3.25 units. ​c) Let ​R(x)=−x2+60x+59 denote the revenue in thousands of dollars generated from the production of x units. What is the​ break-even point?​ (Recall that the​ break-even point is when the revenue is equal to the​ cost.) ​d) Compute and compare the marginal revenue and marginal cost at the​ break-even point. Should the company increase production beyond the​ break-even point?

​a) The marginal cost at production level x=3 is $_____. -Find derivative and plug in 3, $52000 ​b) The cost of producing 3.25 units is ​$____