Differentiation Questions and Answers

12 R t t e 3 t

8 f t 2t

3 f x x 2x ex 4 g x x e

7 g x 3x 1 2x 1 9 V x 2x 3 x 2x I0 Y u u 2 u 3 u 2u 8 f t 2t 4 t

27 30 Find f x and f x 27 f x x e 29 f x x 1 2x 28 f x x5 2ex 30 f x X 3 e

3 26 Differentiate 3 f x x 2x e 5 y X 4 g x x ex ex 1 X 6 y

If r x logs 2x 3x 1 find r x Select the correct answer below In 8 6x 62 2r 3r2 1 O logs 6x2 6x O O 1 In 8 22 3x 1 6r 4 62 In 8 22 3r2 1 2r 3x2 1

If t x ln find t x A 2 1 Select the correct answer below t x 1 2 2 1 O I 2x O t x x t x t x 1 2 O t x 2x x 1 2x x 1 2x x 1

Find the derivative of f x e22 x

d Find the rate of growth after 8 hours Round your answer to three decimal places billion cells per hour e When will the population reach 20 000 cells Round your answer to two decimal places hr Submit Answer 25 Points DETAILS SCALCET8 3 8 010 A sample of a radioactive substance decayed to 96 5 of its original amount after a year Round your answers to two decimal places a What is the half life of the substance yr b How long would it take the sample to decay to 75 of its original amount yr Submit Answer

f x h f x a 5pts Find the average rate of change between x 1 and x 2 2 25 4 2 2 11 4 1 Tim 2 x h 4 x h 2x 4x 470 2 3 2x 2xhth 4 4h 2x 4x impo 2xhth 4h lim 70 h 2xth 4 3 f 2 F 1 X b X a im 2 1 0 4 6 h 0 b 5pts Find the average rate of change between x 1 and x 1 h30 C 5pts Find the instantaneous rate of change at x 1 f x th f x lim h30 2 1 2 1 f b Fla Xb X a 4 1 im 2 xth 4 x h 2x 4x h 0 linn 70 2x2 4 xhth 4x 4h 2x 4X h xhth 4h 7 h 1 h 4 1 0 4 8

Find y and y y y y x In 6x

Let f x cx In cos x For what value of c is f 4 1 C Submit Answer

Derivative of an inverse function The function x x 1 is one to one for x 0 and has an inverse on that interval Find the slope of the curve y f x at the point 3 1 x 2 1 1 3 3 1 4 y f 6 X 9 xy 1 x 10 yz g metb m tan 1 THEOREM 3 21 Derivative of the Inverse Let f be differentiable and have an inverse which f x 0 then is differentia

Derivatives involving the inverse sine Compute the following deriva d dx sin x 1 b cos sin x oof n

If g x ln x 5x find g x 1 24 52 O O g x g x O g x g x 450 4x 5 4x 5 x4 5x 1 4x 5

Let h x f x g x If f x 4x 3x and g x ln 3x 1 what is h x

Let h x f x g x If f x 2x 2x 3 and g x x 3x 4 what is h 1 Provide your answer below h 1

Let f x 5x cos 5x 3 f x

Question 2 Is y 98 10 increasing or decreasing increasing O decreasing Question Help Message instructor

Question 5 You are choosing between two different cell phone plans The first plan charges a rate of 22 cents per minute The second plan charges a monthly fee of 34 95 plus 12 cents per minute How many minutes would you have to use in a month in order for the second plan to be preferable Round up to the nearest whole minute minutes Question Help Video Message instructor

Consider the function f x x2 4x 8 a 5 pts Use the limit definition of derivative to find f x You must show all your work Tim 116 lim X h 4 8 X 4X 8 h 2 X 2th th 48 8 X 4x 8 2 xh th h 140 lim 70 n lim h 0 2X 2 X how b 5 pts Find the x values where the tangent line to f x is horizontal You must show all your work 2 f x 2 4 2 8 0 4 8 8 0 set fc then Solv 4 0 4 F X 4 f a

1 Consider the function f x x 4x 8 a 5 pts Use the limit definition of derivative to find f x You must show all your work

Write your answer in interval notation 2x 4x 0 2 0

The quantity of oxygen that can dissolve in water depends on the temperature of the water So thermal pollution influences the oxygen content of water The graph shows how oxygen solubility S varies as a function of the water temperature T SA mg L 16 12 8 4 8 16 24 32 40 TCC What is the meaning of the derivative S T OS T is the rate at which oxygen solubility changes with respect to the oxygen solubility OS T is the rate at which temperature changes with respect to the temperature OS T is the rate at which oxygen solubility changes with respect to the water temperature OS T is the rate at which temperature changes with respect to the oxygen solubility OS T is the current level of oxygen solubility for a given temperature 0

The cost of producing 2 ounces of gold from a new gold mine is C f x dollars a What is the meaning of the derivative f 2 What are its units Of x is the rate of change of the production cost with respect to the number of ounces of gold produced Its units are dollars per ounce Of x is the average production cost with respect to the number of ounces of gold produced Its units are dollars per ounce Of x is the total production cost Its units are dollars Of x is the total amount of gold produced Its units are ounces b What does the statement f 700 13 mean O The total cost to produce 13 ounces of gold is approximately 700 O After 13 ounces of gold have been produced the average production cost is 700 ounce So the cost of producting 13 ounces is about 700 ounce O After 700 ounces of gold have been produced the average production cost is 13 ounce So the cost of producing 700 ounces is about 13 ounce After 700 ounces of gold have been produced the rate at which the production cost is increasing is 13 ounce So the cost of producing the 700 th or 701 st ounce is about 13 c Do you think the values of f x will increase or decrease in the short term What about the long term Explain In the short term the values of f x will Select an answer of start up costs as a increases But eventually f x might operations because more efficient use is made Select an answer due to large scale

Find f a f t 6t 14 t 6 To find f a we will use the formula f t f a t a f a lim t a Since f t 6t 14 t 6 f a lim t a we have 6t 14 t 6 6a 14 a 6 t a Simplifying everything we get f a lim

Find f a f x 4x 4x 1 f a

Consider the functions f x 9x 1 and g x x 1 a Find f g x b Find g f x c Determine whether the functions f and g are inverses of each other a What is f g x f g x x Simplify your answer Give any values of x that need to be excluded from f g x Select the correct choice and if necessary fill in the box within your choice A x Use a comma to separate answers as needed B No values should be excluded from the domain b What is g f x g f x Simplify your answer

Given f x 6x and g x 4x 6 evaluate the following expressions a fog 4 b gof 2 c fof 1 d gog 0 a fo g 4 420 Simplify your answer b gof 2 582 Simplify your answer c f of 1 Simplify your answer

If f x f x 14 H t dt then

Suppose h x f g x Given the table of values below determine h 1 f x g x f x g x 3 2 7 10 6 4 6 3 5 5 1 in your answer X 1 2 4 5 7

Determine s x if s x ln 4x 5x 2x 5 Select the correct answer below 12x 10x 2 4x 5x 2x 5 O 1 O 4x 5x 2x 5 12x 10x 2 O In 4x3 5x 2x 5 O In 12x 10x 2 4x 5x 2x 5 12r 10x 2

Find the derivative of the function y xe Select the correct answer below O e 2x 1 O e x 2x O e x 1 O 2xe x 2x PLA

Determine s x if s x ln 4x3 5x 2x 5

If t x Select the correct answer below O 0 04e In x O O t x 0 04 what is t x O In x es e t x 0 04 t x In x In ln x e In x In x xex 1 e In x t x 0 04 0 04e 1 I

What is f x if f x 40x p

ive a formula for a piecewise defined function f for the raph shown An too Q OA f x OB f x Oc f x OD f x 2 if x 0 x 6 ifx20 2 ifx 0 x 6 if x 0 2 if x 0 x 6 ifx20 2 if x 0 x if x20

Factor out the smallest power from each expression Wr a 5 3a a

Which is the definition of the derivative of f x 6x 6 x h 6 h f x lim h 0 O f x 6 f x lim h 0 f x lim 6 x h 6hx h f x lim h 0 6x h h 0 h 6 x h 6x h

The position of an object moving on a line is given by s t 6t t t2 0 where s is in metres and t is in seconds a b c Determine a model to represent the velocity Sketch the model for velocity Determine the velocity at t 2

Let h x f x g x If f 6 4 f 6 2 g 6 2 and g 6 5 what is h 6 Do not include h 6 in your answer For example if you found h 6 7 you would enter 7

Let f x 4 x 8 f x lim h 0 Your answer above and the next few answers below will involve the variables h and z The expression inside the limit simplifies to a simple fraction with Numerator Then according to the definition of derivative Denominator NOTE Both of the previous two parts must be correct in order to receive partial credit We can cancel the factor appearing in the denominator against a similar factor appearing in the numerator leaving a simpler fraction with Numerator Denominator NOTE All three of the previous parts must be correct to receive partial credit Taking the limit of this fractional expression as h 0 gives us f x

Find the domain of f x inf 5 U 1 inf U 1 Answers Attempt 4 of 4 0 9x 9 inf A help intervals Answer Enter the solution in interval notation 6

Use the definition of the derivative don t be tempted to take shortcuts to find the derivative of the function f x 3x 4 Then state the domain of the function and the domain of the derivative Note When entering interval notation in WeBWork use I for I for and U for the union symbol If the set is empty enter without the quotation marks x Domain of f x Domain of f x

Let f x x 41 Evaluate the following limits lim 4 1 2 f 4 lim 4 f 2 1 4 Thus the function f x is not differentiablo at 1

13 y x 3 1 x

Find the period y tan 5x Graph the function 10 15 Sn H 4 n 5 5 S 10 10 5 2 Sx 2 1 5 15 3 3 S 10 X 24 2x 3x 10 14 HA A 10 5 101 10 S 10 A 10 32 ff

In Problems 1 3 find the general solution draw a phase portrait and determine whether the equilibrium point at the origin is asymptotically stable stable but not asymptotically stable or unstable 1 dx dt 2 51 4 2 X

Problem 6 Problem 6 Let F x f xf x Suppose that 4 5 4 2 and f 10 5 Find F 2 185