Calculus Questions

Problem 5 Let f x x x Find f 4

Problem 2 If f x x x x 5 and f 1 2 find 1

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

Let I S e e 2x 1 Then for an arbitrary constant c the value of J I equals a log 2 e4x x 1 2x b log 1 c log d log ef e4x e x 1 e x e x e 1 et 1 4x dx J f les e x et 1 e x e 1 2x 4x e le4x x 1 e x 1 c C C C e 4x dx

3 Let f t cos t sin t For each a 4 define Ra to be the region between the graph of f and the t axis and between the vertical lines t 4 and t a Here we label the horizontal axis as the t axis a In the diagram below sketch the region Ra Without using the fundamental theorem of calculus determine whether the rate of change of the signed area of the region is positive for this value of a Explain your answer in terms of how the signed area is changing at this point N b Find the maximum and minimum values of the rate of change of the signed area of Ra for a 4 1 5 31

Between 2006 and 2016 the number of applications for patents N grew by about 4 2 per year That is N t 0 042N t Find the function that satisfies this equation Assume that t 0 corresponds to 2006 when approximately 456 000 patent applications were received Estimate the number of patent applications in 2021 Estimate the rate of change in the number of patent applications in 2021 a b c

The amount of carbon 14 present in animal bones after t years is given by P t Poe 0 00012097t A bone has lost 36 of its carbon 14 How old is the bone To the nearest year what is the age of the bone Round to the nearest integer as needed

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

1 1110 6 The graphs of the function f x given in blue thinner and g x given in red thicker are plotted above Suppose that u x f x g x and v x f x g x Find each of the following u 1 V 1

Suppose that f x 5e ex Find f 3 f 3

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

Find the equation of the tangent line to the curve y 9xe at the point 0 0 The equation of this tangent line can be written in the form y mx b Find m and b m b

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

The following table gives the values for functions f and g and their derivatives for integer values of x between 1 and 5 X 1 2 3 4 4 f x 4 2 f x 5 3 g x 5 4 2 1 g x 3 2 4 3 Let ind p 2 q 3 r 4 p x f x g x q x f x g x st 5 1 1 5 4 1 1 4 r x xf x g x X

if f 5 7 g 5 2 f 5 8 and g 5 5 find the values of the following a fg 5 b f g 5 c g f 5

Let f x 8 x 2 Find f x

Differentiate Tal P a 9

Suppose that and Find h 3 h 3 f 3 10 h x g 3 6 f x g x f 3 1 and g 3 7

By applying the Product Rule twice one can prove that if f g and h are differentiable then fgh f gh fgh fgh Now in the above result letting f g h yields Use this last formula to differentiate y e d x 3 x x

Compute the derivatives of the following functions a The Michaelis Menten kinetics function dv do b The Hill function V y Kx k x Ax an xn

1 Define 3 4 x if 2 x 2 if x 2 f x a Graph f from x 2 to x 8 1 f f x dx using the geometric interpretation of the integral and commonly 2 x 5 b Compute known area formulas Your work should make clear what regions you consider when computing your answer indicate these regions in your diagram for a

Compute the derivative of the given function 12x 1 f x 4x 8x 8 4x 8x

Let f x f x Find f 4 f 4 x 7 Find f x E E

Differentiate the following function R x R x 6 28

6x x Find the derivative of F x Note that you can do this two ways either by simplifying first or by using the Quotient Rule Both approaches should obtain the same answer F x

Differentiate g x g x 1x 2x

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

r Let f x 4x 3 Evaluate f x at the following points A f 3 B f 8

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

Differentiate Y u Y u u u u5 3u

if f x 5x8 4x5 6x 6x find f x and f 1 f x f 1

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

Find a second solution of each differential equation using the reduction of order formula 7 1 2x x y 2 x 1 y 2y 0 Y x 1 1

Solve the rational inequality Express your answer in interval notation x 4 x 4 V X X

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

Suppose that in 1623 a man bought a diamond for 29 Suppose that the man had instead put the 29 in the bank at 3 interest compounded continuously What would that 29 have been worth in 2000 In 2000 the 29 would have been worth Do not round until the final answer Then round to the nearest dollar as needed

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

Find y and y y y y x In 6x 9 6 X 2xIn 6x x 1 2In 6x 2

A 10 foot ladder is leaning against a wall If the top of the ladder slides down the wall at a rate of 2 feet per second how fast is the bottom moving along the ground when the bottom of the ladder is 1 feet from the wall

A tank contains 70 kg of salt and 2000 L of water Pure water enters a tank at the rate 12 L min The solution is mixed and drains from the tank at the rate 6 L min a What is the amount of salt in the tank initially kg amount b Find the amount of salt in the tank after 2 5 hours kg amount c Find the concentration of salt in the solution in the tank as time approaches infinity Assume your tank is large enough to hold all the solution

A drug is administered intravenously at a constant rate of 7 mg hour and is excreted at a rate proportional to the quantity present with constant of proportionality k 0 a Set up a differential equation for the quantity Q in milligrams of the drug in the body at time t hours Your answer will contain the unknown constants r and k Q 0 b Solve this differential equation assuming there is no drug in the body initially Your answer will contain r and k Q c What is the limiting long run value of Q OG 1

The instantaneous rate of change of the value of a certain investment P is proportional to its value That is to say TP If r 7 and P 0 4500 P t

Find the remainder when f x x88 5x 2 is divided by x 1

Find the horizontal H A asymptote and the vertical s asymptotes V A of the function 2x 2 f x x x 2

12 Find the quotient Q x and the remainder R x when g x x Find the sum of all the zeros of the polynomial function g x 4x 0