Application of derivatives Questions and Answers

2 Refer to the function below to do the problems that follow Do not use a calculator That is for each problem be sure to show your work or justify your answer f x x 2x15 2x 3 a State the domain Use interval notation 4 points b Give the equation of each VERTICAL asymptote 4 points

1 Refer to the graph below to do the problems that follow 10 6 6 8 10 a State the domain of the function whose graph is shown Use interval notation 4 points b Give the equation of each VERTICAL asymptote 4 points c Give the equation of the HORIZONTAL asymptote

1 Given a function of g x 10 3x 5 answer the following questions 2 points each A Find the slope of the line tangent to the graph of g x at 0 2 B Determine an equation of the tangent line at 0 2

Given a graph shown to the right answer the following questions A Find the values of x in 0 4 where g x is discontinuous 2 points B Find the values of x in 0 4 where g x is indifferentiable 2 points C Draw g x on the graph below 2 points the den Moed 13 16 g x 1 1 0 1 2 3 YA 4 3 2 1 0 A 2 y g x 2 173 3 4

Find the derivatives of the following functions 2 points each Deniwalotarll A f t 10 t 500 t 1000 79 5x 1 3 nex

The function graphed below is defined by a polynomial expression of degree 4 Use the graph to solve the exercise YA M a If f is increasing on an interval then the y values of the points on the graph rise b Iff is decreasing on an interval then the y values of the points on the graph fall V as the x values increase From the graph of f we see that f is increasing on the intervals as the x values increase From the graph of f shown above we see that fis decreasing on the intervals x Enter your answer using interval notation X Enter your answer using interval notations

Find the first and second derivatives of the following function f x 9x 3x 6x x f x

Differentiate f x 428 5 cos x f x

You would like to have 3500 in 5 years for a special vacation following graduation by making deposits at the end of every six months in an annuity that pays 4 comp parts a and b 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 5 years you should deposit at the end of every six months Do not round until the final answer Then round up to the nearest dollar 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 P 1 1 n P A 2 M n X

he given X 7 X 49 f x Tipiones Use integers or tractions for any numbers in the equations OC There are no vertical asymptotes Find the horizontal or oblique asymptotes Select the correct choice below and if necessary fill in the answer box to complete your choice A The equation of the oblique asymptote is Use integers or fractions for any numbers in the equation B The equation of the horizontal asymptote is y 0 Use integers or fractions for any numbers in the equation OC There are no horizontal or oblique asymptotes Find the coordinates of any holes in its graph Select the correct choice below and if necessary fill in the answer box to complete your choice OA There is a hole in the graph off at the point Type an ordered pair using integers or fractions B There is no hole in the graph of t

a Find the average rate of change of the area of a circle with respect to its radius r as r changes from 4 to each of the following 1 4 to 5 ii 4 to 4 5 iii 4 to 4 1 b Find the instantaneous rate of change when r 4 A 4

The gas law for an ideal gas at absolute temperature T in kelvins pressure P in atmospheres and volume V in liters is PV nRT where n is the number of moles of the gas and R 0 0821 is the gas constant Suppose that at a certain instant P 9 0 a increasing at a rate of 0 15 atm min and V 12 L and is decreasing at a rate of 0 16 L min Find the rate of change of T with respect to time in K min at that instant if n 10 mol Round your answer to four decimal places dT dt K min

a lim f x X 2 b lim f x X 0 a Find lim f x or state that it doesn t exist Select the correct choice below and fill in any answer X 2 boxes in your choice OA lim f x X 2 Type an integer OB The limit does not exist b Find lim f x or state that it doesn t exist Select the correct choice below and fill in any answer boxes X 0 in your choice OA lim f x X 0 Type an integer OB The limit does not exist

Consider a generic quadratic function f x ax b c with characteristics described below Determine the domain and range of f The vertex of the graph of is 24 6 a 0 The domain of is The range of f is

Differentiate the function f x sin 9 In x I 9 cos 9In x f x cos X

y x8x y x 8e8x ln x 8eln x in x Enhanced Feedback Please try again First take the logarithm on both sides to get an expression for In y Second differenti when logarithmic differentiation is used either the product rule or quotient rule must be applied in the se Need Help Submit Answer 0 12 Points Read It DETAILS Find y if y In 7x 9y Watch It PREVIOUS ANSWERS SCALCET9 3 6 057

Find and simplify the difference quotient f x 7 OA 1 OB 0 O C 1 OD 7 h f x h f x h 11 h 0 for the given function

Explain how the graph of the function f x give the domain and range To obtain the graph of f shift the graph of y down Graph the function f x OA 1 x 4 Q X x 4 B 3 can be obtained from the graph of y Then graph fand to the left 4 units reflect across the x axis and shift 3 units 3 Choose the correct graph below Q o o S O D oo S

lain how the graph of the function f x 2 can be obtained from the graph of y Then graph f and give main and range To obtain the graph of f shift the graph of y SLEER up to the left unit s to the right down

Charlotte Mecklenburg Schools 7 Noah is helping to collect the entry fees at his school s sports game Student entry costs 2 75 each and adult entry costs 5 25 each At the end of the game Diego collected 281 25 Select all equations that could represent the relationship between the number of students s the number of adults a and the dollar amount received at the game a 281 255 25a 2 75s 281 25 2 758 5 25 b a c 281 25 5 25s a d 281 25 2 75a s e 281 25 5 25s a

What is the largest open interval over which f x increases decreases and is constant What is the largest open interval over which f x increases Select the correct choice below and if necessary fill in the answer box to complete your choice OA The open interval is Type your answer in interval notation Use a comma to separate answers as needed B None What is the largest open interval over which f x decreases Select the correct choice below and if necessary fill in the answer box to complete your choice The open interval s is are Type your answer in interval notation Use a comma to separate answers as needed B None OA

What is the equation of the vertical asymptote of the graph of y horizontal asymptote 1 What is the vertical asymptote of the graph of y X 4 The equation of the vertical asymptote is x 5 1 X 4 5 What is the equation of the

feet of fencing find the dimensions that maximize the area of the pen using all of the available fencing material Pen 1 Let w be the width of the enclosure and let l be the length of the enclosure as shown above Write an function for the area A of the enclosure in terms of w A w What width w and length I would maximize the area W Barn W ft and l What is the maximum area square feet feet W

What is the domain of x What is its range The domain of f x Type your answer in interval notation

K For f x a fog 6 X 5 Type your answer in interval notation Use integers or tractions for any numbers in the expression Sim your answer 5 and g x find the following composite functions and state the domain of each X b gof c fof d gog 5X 25 6 Simplify your answer Use integers or fractions for any numbers in the expression b gof x The domain of gofis 5 U 5 00 Type your answer in interval notation Use integers or fractions for any numbers in the expression Sin your answer 6 x 5 31 5x Simplify your answer Use integers or fractions for any numbers in the expression c f of x 31 31 7 51 0 5 The domain off of is o 5 U 5 Type your answer in interval notation Use integers or fractions for any numbers in the expression Sir your answer d gog x Simplify your answer Use integers or fractions for any numbers in the expression

Determine the slope of the tangent to the graph at x 4 Determine Y 8 dy dx for Determine the slope of the tangent to the graph at x 4 0 y 16x 13 Given that f x lim f x h f x h h 0 Y Give answers rounded to two decimal places 12 5 Determine the slope of the tangent to the graph at x 5 5 X when x 6 11 22 y 5x4 x 1 for all x evaluate Determine the equation of the tangent line to the curve at point 1 6 y x 5x 2x

The derivative of a function f in terms of a limit is defined by which of the following lim f x h f x h 0 h lim x h lim f h f r h h 0 lim lim h 0 f x h f x h Determine Determine f x f a a y 3x 18 5 dy dr dy dr fi 0 for What value is the following limit calculating 4 h 4 for fi 4 for f x x fi 16 for f x x fi 4 for f x x f a A y 732

A curve passes through the point 0 7 and has the property that the slope of the curve at every point P is twice the y coordinat of P What is the equation of the curve Use x as the independent variable y x

A fireplace arch is to be constructed in the form of a semiellipse The opening is to have a height of 3 feet at the center and a width of 8 feet along the base The contractor cuts a string of a certain length and nails each end of the string at the foci of the base in order to sketch the outline of the semiellipse a What is the total length of the string Select b How far from the center should the string be nailed into the base Select

Differentiate the function F t F t In 5t 1 4 2t 1 5

If f x In 4 e5x find f 0 f 0

Find y and y y y y In sec 2x tan 2x

A satellite dish is the shape of a paraboloid The dish is 38 inches wide and 10 inches deep How many inches should the receiver be located from the vertex for optimal reception round to the nearest thousandth Receiver

Determine the location of the focus and the directrix of the parabola y 6y 8x 23 0 Focus Select Directrix Select Select

The annual consumption of beef per person was about 64 5 lb in 2000 and about 60 3 lb in 2008 Assume B t the annual beef consumption t years after 2000 is decreasing according to the exponentia decay model a Find the value of k and write the equation b Estimate the consumption of beef in 2010 c In what year theoretically will the consumption of beef be 20 lb a Select the correct answer below and fill in the answer box to complete your choice Round to four decimal places as needed 60 3 O A O C B t B t 64 5 60 3e OB B t 64 5e OD B t 60 3e

Problem 1 Find the value of y when x 0 if xy ey 7 e

Problem 3 Let f x ln 1 g x where g 8 x 8 Do not include y in your answer e5 1 and g 8 2es Find the equation of the tangent line to fat

Find the parabola with equation y ax bx whose tangent line at 1 2 has equation y 4x 2 a E b

Differentiate F y 9

Consider f x 14 e A Find the slope of the graph of f x at the point where the graph crosses the x axis slope B Find the equation of the tangent line to the curve at this point 1 C Find the equation of the line perpendicular to the tangent line at this point This is the normal line

if f x x5 5e find f 4 f 4 Use this to find the equation of the tangent line to the curve y tangent line can be written in the form y mx b Find m and b m 5 5e at the point a f a when a 4 The equation of

Let f x 6e e 0 B

Find the equation of the tangent line to the curve y x x at the point 9 27 y

For what value s of x is the tangent line of the graph of parallel to the line y 2x 1 7 3 f x 8x 12x 50x 72 If there is more than one value enter your answer as a comma separated list eg 2 3 4 without quotes

Let f x 3x 5x 3 Compute f 3 Answer Use this to find the equation of the tangent line to the parabola y 3x 5x 3 at the point 3 45 The equation of this tangent line can be written in the form y mx b Determine m and b m

2 Define g to be the integral function g x where f is the function graphed below 2 omputo al 1 Tuotify Lancy f t dt x sin x your work f

Suppose that Po is invested in a savings account in which interest is compounded continuously at 6 9 per year That is the balance P grows at the rate given by the following equation dP dt 0 069P t a Find the function P t that satisfies the equation Write it in terms of Po and 0 069 b Suppose that 1500 is invested What is the balance after 1 years c When will an investment of 1500 double itself a Choose the correct answer below OA P t 0 069Poet OB Po P t e 0 0691 OC P t P t e 0 069t OD P t Poe0 069t b The balance after 1 year is Type an integer or decimal rounded to two decimal places as needed c The doubling time is year Type an integer or decimal rounded to two decimal places as needed

How much money must you invest now at 4 7 interest compounded continuously in order to have 10 000 at the end of 6 yea You must invest Round to the nearest cent as needed

An athlete signs a contract that guarantees a 11 million salary 5 yr from now Assuming that money can be invested at 6 9 with interest compounded continuously what is the present value of that year s salary Round to the nearest dollar as needed Do not include the symbol in your answer

The size of a certain insect population is given by P t 500e 03t where t is measured in days At what time will the population equal 200 It will take days for the population to equal 2000 Round to one decimal place as needed