Application of derivatives Questions and Answers

A thermometer reading 12 C is brought into a room with a constant temperature of 25 C If the thermometer reads 17 C after 4 minutes what will it read after being in the room for 7 minutes Assume the cooling follows Newton s Law of Cooling U T Uo T ekt Round your answer to two decimal places 24 57 C 8 77 C 30 56 C 19 44 C QUESTION 20 rabbit

QUESTION 19 Solve the problem A thermometer reading 12 C is brought into a room with a constant temperature of 25 C If the thermometer reads 17 C after 4 minutes what will it read after being in the room for 7 minutes Assume the cooling follows Newton s Law of Cooling U T Uo T ekt Round your answer to two decimal places 24 57 C 8 77 C 30 56 C 19 44 C QUESTION 20

Find the equation of the line tangent to the graph of f x at the indicated x value f x 3 sin 5x x 0 Tangent line y

18 QUESTION 18 Solve the problem A fossilized leaf contains 35 of its normal amount of carbon 14 How old is the fossil to the nearest year Use 5600 years as the half life of carbon 14 28 672 3474 8466 33 664 19 20

If x y 90 and 9 find when y 3 and x 9

QUESTION 15 Solve the equation log3 x log3 x 24 4 No real solutions 27 O 53 O 3 27

The graph of a logarithmic function is shown Select the function which matches the graph Oy log 1 x Oy log x 1 Oy 1 log x

QUESTION 21 The function f is one to one Find its inverse f x 6x 7 x 7 f x x 7 f1 x 2 6 7 f1 x 2 7 O O f x x 7

QUESTION 20 Solve the problem Conservationists tagged 60 black nosed rabbits in a national forest in 1990 In 1993 they tagged 120 black nosed rabbits in the same range If the rabbit population follows the exponential law how many rabbits will be in the range 10 years from 1990 1210 605 74 148 QUESTION 21

Find the power function that the graph of f resembles for large values of Ix f x x x 9 x y x7 y x7 y x3 Oy x QUESTION

Form a polynomial f x with real coefficients having the given degree and zeros Degree 3 zeros 3 and 3 2i Of x x3 x2 5x 39 f x x 3x 5x 39 f x x3 x2 11x 39 Of x x3 3x 5x 52

Find the domain of the rational function 2x 4 3x2 6x 45 f x x x 3 x 3 x 5 x x 3 x 5 x x 3 x 5 all real numbers

Find the domain of the composite function f g 0000 f x g f x g x 63 X 9 x x 0 x 9 x 7 x x 0 x 7 x x is any real number x x 0 x 9

05 Find the equation of the tangent line to the graph of f x when z 0

Here is a graph of the function g V 5 4 3 60 4 w 24 1 H 24 3 4 y 4 fo 5 X Use the graph to find the following If there is more than one answer separate them with commas a All values at which g has a local maximum b All local maximum values of g

be an absolute minimum value for f x on the interval 1 4 34 Find the absolute maximum of f x x4 8r on the interval 0 7 11 B

concave down and find any r values of inflection points 32 Find the x values of the inflection points of f x x 18x

A parcel delivery service has contracted you to design a closed box with a square base that has a volume of 11 000 cubic inches Complete parts a d M A X X a

lath 126 3 12 pts a What is the average rate of change of the function f x 1 the equation of the secant line through 3 f 3 and 1 2 f 1 2 reduced fractions c Sketch a graph of f and the secant line from 3 to 1 2 b What i Write your answers in terms

Find all real zeros of the function f x 6x 19x 45x 100 Given 4 is a factor of f x Enter the real zeros separated by commas Give as exact values Question Help Mec

The half life of cesium 137 is 30 years Suppose we have a 180 mg sample a Find the mass in mg that remains after t years y t b How much of the sample in mg remains after 80 years Round your answer to two decimal places X mg c After how many years will only 1 mg remain Round your answer to one decimal place yr t mg Enter a number AWA

a The domain of f x 2 4 b The range of f x 1 3 2 1 The graph of y f x given above starts at x 2 and ends at x 4 Use the graph of y f x to identify a 2 473 c f 0 I d For what values of x does f x 1 Doesn t wom e On what interval s is f x 0 2 4 f How many 3 3 2 2 4 1 b 1 33 d A e nes does the line y intersect the graph of f x f Q4 and

1a b C Evaluate f 3 for f x 3x 7x 4 6 Find the third derivative of f x 3x 7x 4 Find the point on the parabola y x 3x 4 where the slope of the tangent is 4 If V represent the volume of an object what does represent d DATOSESI ORCHAEOL C4 3

9 O O O D On what interval s is f increasing and on what interval s is f decreasing Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The function is increasing on The function is never decreasing Simplify your answer Type your answer in interval notation Type an exact answer using radicals as needed Use a comma to separate answers as needed OB The function is decreasing on The function is never increasing Simplify your answer Type your answer in interval notation Type an exact answer using radicals as needed Use a comma to separate answers as needed OC The function is increasing on and decreasing on Simplify your answers Type your answers in interval notation Type exact answers using radicals as needed Use a comma to separate answers as needed OD The function is never increasing or decreasing Determine the coordinates of the relative extrema Select the correct choice below and if necessary fill in the answer box to complete your choice O Sketch the graph of the following function Indicate where the function is increasing or decreasing where any relative extrema occur where asymptotes occur where the graph is concave up of conca O f x 2 x 7 OA The coordinates of the relative extrema are Type an ordered pair Type an exact answer using radicals as needed Use a comma to separate answers as needed OB There are no relative extrema Determine the vertical asymptote s Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The function has one vertical asymptote Type an equation OB The function has two vertical asymptotes The leftmost asymptote is Type equations OC The function has no vertical asymptotes Determine the horizontal asymptote s Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The function has two horizontal asymptotes The top asymptote is and the bottom asymptote is Type equations OB The function has one horizontal asymptote Type an equation OC The function has no horizontal asymptotes Determine the slant asymptote s Select the correct choice below and if necessary fill in the answer box to complete your choice OA The stant asymptote s is are Type an equation Use a comma to separate answers as needed OB The function has no slant asymptotes and the rightmost asymptote is

Sketch the graph of the following function Indicate where the function is increasing or decreasing where any relative extrema occur where asymptotes occur where the graph is concave up or concave down where any points of inflection occur and where any intercepts occur f x 2 x 7 On what interval s is f concave up and on what interval s is f concave down Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The function is concave up on and concave down on Simplify your answers Type your OB The function is concave up on Simplify your answer Type your OC The function is concave down on and is never concave up Simplify your answer Type your answer in interval notation Use a comma to separate answers as needed Determine the coordinates of the point s of inflection Select the correct choice below and if necessary fill in the answer box to complete your choice answers in interval notation Use a comma to separate answers as needed and is never concave down answer in interval notation Use a comma to separate answers as needed OA The coordinates of the point s of inflection are Simplify your answer Type an ordered pair Use a comma to separate answers as needed OB There are no points of inflection Determine the x intercept s Select the correct choice below and if necessary fill in the answer box to complete your choice OA The x intercept s is are Type an ordered pair Type an exact answer using radicals as needed Use a comma to separate answers as needed OB There are no x intercepts Determine the y intercept s Select the correct choice below and if necessary fill in the answer box to complete your choice OA The y intercept is Simplify your answer Type an ordered pair Use integers or fractions for any numbers in the expression OB There is no y intercept Sketch the graph of the function Choose the correct graph below OA AY Q OB Ay GILD Q OC Q O D Q

For the following function a give the coordinates of any critical points and classify each point as a relative maximum a relative minimum or neither b identify intervals where the function is increasing or decreasing c give the coordinates of any points of inflection d identify intervals wher up or concave down and e sketch the graph h x 2x 24x a What are the coordinates of the relative extrema Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The relative minimum point s is are and the relative maximum point s is are Simplify your answers Use integers or fractions for any numbers in the expression Type an ordered pair Use a comma to separate answers as needed OB The relative maximum point s is are and there are no relative minimum point s Simplify your answer Use integers or fractions for any numbers in the expression Type an ordered pair Use a comma to separate answers as needed and there are no relative maximum point s OC The relative minimum point s is are Simplify your answer Use integers or fractions for any numbers in the expression Type an ordered pair Use a comma to separate answers as needed OD There are no relative minimum points and there are no relative maximum points b On what interval s is h increasing or decreasing Select the correct choice below and if necessary fill in the answer box es to complete your choice A The function is increasing on The function is decreasing on Simplify your answers Type your answers in interval notation Use a comma to separate answers as needed OB The function is decreasing on The function is never increasing Simplify your answer Type your answer in interval notation Use a comma to separate answers as needed OC The function is increasing on The function is never decreasing Simplify your answer Type your answer in interval notation Use a comma to separate answers as needed OD The function is never increasing or decreasing c What are the coordinates of the inflection point s Select the correct choice below and if necessary fill in the answer box to complete your choice OA The coordinates of the inflection point s are Simplify your answer Type an ordered pair Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed OB There are no inflection points d On what interval s is h concave up or concave down Select the correct choice below and fill in the answer box es to complete your choice Simplify your answer Type your answer in interval notation Use a comma to separate answers as needed OA The function is concave up on and concave down on OB The function is concave up on OC The function is concave down on GIIB and is never concave down and is never concave up

Determine the horizontal asymptote of the function If none exists state that fact 4x 5x 8x X Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The function has one horizontal asymptote Type an equation OB The function has two horizontal asymptotes The top asymptote is Type equations OC The function has no horizontal asymptotes and the bottom asymptote is

For f x 2 x 2x 3 x 2 determine the x intercepts and the end behavior Enter intercepts as a comma separated list of points Intercepts help points As x f x As x o f x help numbers help numbers

83 An office lamp manufacturer has weekly revenue and cost equations both in dollars given by R 38x 0 1 x and C 5x 2000 respectively where x is the number of lamps made and sold each week If the number of lamps made and sold is increasing at a rate of 30 lamps per week find the rate of change of the company s profit with respect to time when 200 lamps are made and sold each week Interpret your answer

Use the graph shown to find the following a The domain and range of the function b The intercepts if any c Horizontal asymptotes if any d Vertical asymptotes if any e Oblique asymptotes if any 9 6 3 A Q M What is the range Select the correct choice below and fill in any answer boxes within your choice OA The range of the function is y Type an inequality in the form y Use integers or fractions for any numbers in the expression Use comma to separate answers as needed

Determine the difference quotient S x h f x h where h 0 for each of the following function 3 x n n

Select all of the following functions for which the extreme value theorem guarantees the existence of an absolute maximum and minimum Select all that apply Of x ln 1 x over 0 2 Og x ln 1 x over 0 2 h x x 1 over 1 4 0 k x over 1 4 None of the above

olve the initial value problem 1 2 A s sint cost 2 C s 2sint 1 21 cost sin t s 3 B s sin t cost 2 D s sin t cost 4

Find the critical numbers of the following function q x 4x x 4x 1 Select the correct answer below 0 A O and 01 02 O There are no critical numbers

Analyze the following graph of f x Select all of the intervals over which the graph of f x is concave up Select all that apply 0 1 1 1 1 00 None of the above T 24

5 4 3 2 2 1 1 2 3 4 X The function graphed above is decreasing on the interva 0 The inflection point is at x

1 The function f x 8x critical number X 2 3 X Question Help Video Add Work 2 e 42 has one critical number Algebrically use the derivative to find the

Mark the critical points on the following graph 36 30 Clear All 5 4 3 2 1 Add Work 18 12 Draw 0 6 12 18 z 2 7 Dot 3

Find the inflection point for the function shown below If there are more than one be sure to spearate them by using a comma If there is not an inflection point record DNE in the answer box If necessary round all numbers to two decimal places f x 4x 66x 216x 2 1

The function f x 2x 33x 108x find the critical numbers The smaller one is x and the larger one is x Question Help Video POEMENE 2 has two critical numbers Use derivatives to algebraically

An ellipse or hyperbola uses the general form Ax Cy Dz Ey F 0 Solving for 5 unknowns A B C D E F requires 5 equations needs 5 points given But if one of the coefficients is divided out A or C then only 4 coefficients remain and only 4 points are needed x Cy Dr Ey F 0 Given 4 points on a vertical ellipse 1 2 517 0 1 933 1 0 65 and 4 449 0 Select the missing coefficients answers have been rounded to the nearest tenth x y Select Select Select y Select 0 x

Algebraically find the critical numbers of the function f x 4x5 5x 20x 7 and classify them Leave answers in 4 decimal places X X X Add Work is a Neither a max or min is a Neither a max or min is a Neither a max or min

Question 2 6 5 7 4 d The function graphed above has an inflection point Add Work

HER PRACTICE ANOTH Of the infinitely many lines that are tangent to the curve y 8 sin x and pass through the origin there is one that has the largest slope Use Newton s method to find slope of that line correct to six decimal places 10 083850

Find the area of the region bounded by the polar curve r 6e on the interval 2T Answer Granh the curve on a calculator Turn in a sketch of the curve

You would like to have 3500 in 3 years for a special vacation following graduation by making deposits at the end of every six months in an annuity that pays 4 compounded semiannually Answer parts a and b i Click the icon to view some finance formulas a How much should you deposit at the end of every six months In order to have 3500 in 3 years you should deposit s at the end of every six months Do not round until the final answer Then round up to the nearest dollar b How much of the 3500 comes from deposits and how much comes from earnings S of the 3500 comes from your deposits and comes from interest Use the answer from part a to find this answer Round to the nearest dollar as needed More info In the provided formulas P is the deposit made at the end of each compounding period for an annuity that pays an annual interest rate r in decimal form compounded n times per year and A is the value of the annuity after t years A 9 r nt auon by mar 1 P A rnt A

K Analyze the polynomial function f x x x 5 using parts a through h below The x intercept s is are 0 5 Simplify your answer Type an integer or a fraction Use a comma to separate answers as needed Type each answer only once The y intercept s is are 0 Simplify your answer Type an integer or a fraction Use a comma to separate answers as needed Type each answer only once c Determine the real zeros of the function and their multiplicity Use this information to determine whether the graph crosses or touches the x axis at each x intercept 4 The real zero s of f is are 0 5 Simplify your answer Type an integer or a fraction Use a comma to separate answers as needed Type each answer only once The lesser zero of the function is of multiplicity 2 so the graph off touches the x axis at x 0 The greater zero of the function is of multiplicity 1 so the graph of f crosses the x axis at x d Use a graphing utility to graph the function The graphs are shown in the viewing window Xmin 20 Xmax 20 Xscl 2 Ymin 30 Y max 30 Yscl 3 Choose the correct graph below A oo OB O C 2 e Approximate the turning points of the graph The turning point s of the graph is are Type an ordered pair Round each coordinate to the nearest hundredth as needed Use a comma to separate answers as needed OD Q

Question 6 The polynomial of degree 3 P x has a root of multiplicity 2 at x 3 and a root of multiplicity 1 at x 5 The y intercept is y 18 Find a formula for P x P x Question Help Video Message instructor

At the time of her grandson s birth a grandmother deposits 10 000 in an account that pays 3 5 compounded monthly What will be the value of the account at the child s twenty first birthday assuming that no other deposits or withdrawals are made during this period Click the icon to view some finance formulas The value of the account will be Round to the nearest dollar as needed

Use technology to find points and then graph the function y 2 log x 2 2 following the instructions below done plotting points Plot at least four points with integer coordinates that fit on the axes below Click a point to delete it 74 S