Application of derivatives Questions and Answers

I 4 59 Sketch a graph of a function that satisfies all the conditions that follow lim f x 2 lim f x 4 f 2 1 lim f x 60 on 2 4 lim f x x f 0 undefined 2111 0

If 0 a 5T find the exact value of each expression below cos e 0 b cos 20 c cos 8 0 3 16 X G

60 Sketch a graph of a function that satisfies all the conditions that follow lim f x 3 lim f x 0 f 0 4 lim f x 1 lim f x 1 f 2 0 lim f x 2 2 0 2 0 I 2 1 2

he profit function in millions for a beverage company for the years 2010 through 2016 can be approximated by f x 26x 780x 4879 where x 10 corresponds to the year 2010 a During what year did a local maximum profit occur What was the maximum profit a A local extremum occurs at a critical number of f Because the derivative exists for every x the only critical number s occur where the derivative is zero Find the derivative of f x local maximum profit occurred in the year o The maximum profit was million

h t COS T cos t Step 3 As previously noted the given function f t 6t sin nt is the product of two differentiable functions of the form f t g t h t where g t 6t and h t sin t We determi h t cos nt So the last thing we must do before applying the product rule is to find g t g t 6t g t We now apply the product rule g t h t g t h t h t g t 6t os nt sin t Therefore we have the following result f t

Evaluate f c at the point f x In 4x 3 c 1 OA f 1 16 OB f 1 0 OC f 1 4 OD f 1 1

We are given the function f x sin 8 In x To write this is the form F x h g x we can let g x 8 In x and h x sin x Step 2 Before applying the Chain Rule we will first find the derivatives g x and h x g x 8 In x g x sin x h x sin x h x

Find the derivative of the function f t 6t sin nt Step 1 We note that the given function f t 6t sin nt is the product of two differentiable functions of the form f t g t h t where g t 6t and h t sin t However before we can apply the product rule we must first find h t Doing so requires the use of the chain rule because h t sin nt is a composite function with u it and h u sin u Furthermore we recall that in general if y sin u where u is a differentiable function of t then by the chain rule we have the following So we first find U It du dt Step 2 70 dy dy du du dt dt itor du dt cos u COC Tt 70 du We let u t and we determined that dt Applying these to the result from the chain rule for y sin u gives us the following dy cos u cos nt du dt dt Therefore we have the following h t sin at du dt TU X 15 Vo 0

Find the equation of the tangent line to the graph off at the given point f x In 1 7x7 at 0 0 The equation of the tangent line to the graph off at the given point is Use integers or fractions for any numbers in the count

or the function shown in the graph list the intervals on which the function is increasing the tervals on which it is decreasing and the location of all local extrema 10 A List the open interval s on which the function is increasing Select the correct choice below and if necessary fill in the answer box to complete your choice OA The function is increasing on the interval s Type your answer in interval notation Use a comma to separate answers as needed B The function is never increasing List the open interval s on which the function is decreasing Select the correct choice below and if necessary fill in the answer box to complete your choice OA The function is decreasing on the interval s Type your answer in interval notation Use a comma to separate answers as needed OB The function is never decreasing Identify any local extrema Select the correct answer below and if necessary fill in any answer box es within your choice OA There is no local minimum and the local maximum maxima occur s at x Use a comma to separate answers as needed OB There is no local maximum and the local minimum minima occur s at x Use a comma to separate answers as needed OC There is are local maximum maxima at x local minimum minima at x Use a comma to separate answers as needed D The function has no local extrema and there is are

For the function below find a the critical numbers b the open intervals where the function is increasing and c the open intervals where it is decreasing f x 2x 3x 36x 18 a Find the critical number s Select the correct answer below and if necessary fill in the answer box within your choice OA The critical number s is are Type an integer or a simplified fraction Use a comma to separate answers as needed OB There are no critical numbers b List any interval s on which the function is increasing Select the correct answer below and if necessary fill in the answer box within your choice OA The function is increasing on the interval s Type your answer in interval notation Use a comma to separate answers as needed OB The function is never increasing c List any interval s on which the function is decreasing Select the correct answer below and if necessary fill in the answer box within your choice OA The function is decreasing on the interval s Type your answer in interval notation Use a comma to separate answers as needed OB The function is never decreasing

Use the quotient rule to find the derivative of the function 2x 2 1 X 5 dy y

Step 2 du We let u nt and we determined that I dt Applying these to the result from the chain rule for y sin u gives us the following dy cos u du dt dt h t cos nt Therefore we have the following h t sin nt os nt 6 du Tdt Submit Skip you cannot come back Need Help Read It X

The differentiable functions f and g are defined for all real numbers x Values of f f g and g for various values of x are given in the table x f x f x g x g x 3 4 1 2 3 1 7 5 7 a If h x f g x find h 3 h 3 b If H x g f x find H 2 H 2 2 8 2 6 7 9 2

Consider the following function 5 1 8t 1 Simplify by rewriting F t using a negative exponent and no fractions F t F t Find the derivative of the simplified function F t

10 Give the accurate definition of a concave upward and a concave downward curve Illustrate each case with a picture

f x f a 2 a for derivative of 5 Explain precisely using a picture where the formula f a lim I a a function f x at a number x a comes from What is the meaning of derivative

218 14 lim n n 32 n 3 n 36 n 3 n n 32n

21 f x 24x7 sin x 4 sec r tan r f 0

1 p 286 10 Minimize c 0 4x 0 1y subject to 30x 0 1 0 2x 0 3y x 0 y 0 20y 0 4y 0 4y IV 600 4 4 5

3 Find the volume of the solid generated by revolving the region bounded by y x and th lines y 1 x 5 about the line y 1

Find the y intercept and slope of the lin y 2 3

Suppose that f x y x y xy E Then the minimum is

Consider the Cobb Douglas Production function P L K 16L0 8 K0 2 Find the total units of production when L 15 units of labor and K 13 units of capital are invested Give your answer to three 3 decimal places if necessary Production units

Evaluate the limit lim h 0 Type your solution in the space provided along with sufficient dtails showing what you did in each step 3 h 27 h

The graph of f x 2x 15x 84x 21 has two horizontal tangents What is the negative value of a where a horizontal tangent occurs What is the positive value of where a horizontal tangent occurs

Evaluate the limit Type your solution in the space provided and include details about what you did in each step lim x 3 sin x 3 x 3

Given 8 3 and x 9 find all x such that the distance between these two points is 13 Separate multiple answers with a comma Answer M

Find the perimeter of the triangle whose vertices are 4 1 3 0 and 1 4 Write the exact answer Do not round 4 Answer

Given f x y 100x 70y 3x 4y xy 2 Evaluate f 6 8 f 6 8

f ftt Let g x f t dt where f is the function whose graph is shown 1 9 6 3 3 y 4 5 3 g 15 22 5 g 18 0 6 9 12 a Evaluate g x for x 0 3 6 9 12 15 and 18 g 0 0 g 3 g 6 0 g 9 40 5 g 12 36 XXXX x x x f x 15 18 21 T9 5 3 002

Write a in the form a T aN N without finding T and N r t c cos t i csin t j dtk

Nm kg Find the length of the major axis of Earth s orbit using Kepler s third law and the fact that Earth s orbital period is 365 256 days Use G 6 6726 10 11 that 1 N 1 kg m s The length of the major axis of Earth s orbit is km M 1 99 x 1030 kg and x 3 14 Re

the speedometer on your car reads a steady 35 mph Could you be accelerating Explain hoose the correct answer and explanation A Yes On a curved path the velocity always changes in magnitude but it may or may not change in direction which means that the acceleration of an object moving on a curved path can zero B No If the car is moving along a curved path x 0 then ay x v 0 and a a T aN 0 OC No The car is not accelerating if the speed is constant D Yes If the car is moving along a curved path x0 then a x v 0 and a a T aN 0

Evaluate the integral 3 2 for 1 31 x 2 dx

Use part one of the fundamental theorem of calculus to find the derivative of the function 1 h x h x 4 In t dt

Find the general indefinite integral Use C for the constant of integration 1 sin 2x dx sin x

120 If oil leaks from a tank at a rate of r t gallons per minute at time t what does 120 the change in the rate of oil that leaks per minute O the amount of time it would take for 120 gallons of oil to leak from the tank O the volume of oil in the tank after oil leaks for 120 minutes r t dt represent the number of gallons of oil that leaked from the tank in the first 120 minutes O the amount of time it would take for the tank to have 120 gallons of oil left over after leaking

80 Suppose the demand function for x thousand of a certain item is p 100 where x 1 and p is in dollars In x a Find the marginal revenue b Find the revenue from the next thousand items at a demand of 9000 x 9 80 80 a The marginal revenue function is R x 100 Inx Inx Use integers or fractions for any numbers in the expression b The revenue from the next thousand items when x 9 is Type an integer or decimal rounded to two decimal places as needed

V 5 20 The volume of a cube with sides of length s is given by Find the rate of change of volume with respect to s when s 13 centimeters a 6 591 cm 12 b 2 197 cm c 169 cm d 507 cm e 338 cm

18 Determine all values of x if any at which the graph of the function has a horizontal tangent y x x 6x 3 x 0 b x 4 c x 0 d x 0 x 4 and e The graph has no horizontal tangents a and X 4

The life span of a certain insect in days is uniformly distributed over the interval 16 28 A What is the expected life of this insect B Find the probability that one of these insects randomly selected lives longer than 23 days A The expected life of this insect is days B The probability that one of these insects randomly selected lives longer than 23 days is Round to four decimal pla

Find the derivative of the following function y 85x 1 dy dx

11 Find all values of c such that fis continuous on f x 4 x x c 5x x c 1 a b c 0 c 5 41 2 d 5 41 5 41 2 2 e 5 41 2 5 41 2

the following differential equation find the integrating factor the general solution and the particular solution satisfying the given initial condition y y 42x6ex y 0 15 The integrating factor is 1 x ex The general solution is y 6x7 C ex The particular solution is y 6x7 15

5 Suppose that x lim f x g x a 11 b 16 C lim f x 5 16 d 5 e 55 lim g x 11 and x Find the following limit

4 Let f x 5x 2 lim g f x x 4 a 1 264 b 256 c 1 282 d 234 25 6 e 16 and g x x4 Find the limit

27 Consider the following a Find an approximation to the integral using a Riemann sum with right endpoints and n 8 R8 9 8 7 x 2x b Draw a diagram to illustrate the approximation in part a y 6 5 4 3 2 2 0 34 2x dx 2 5 c Consider the following theorem 3 Use this to evaluate the integral If f is integrable on a b then 7 2 4 9 2 en 1 x dx 9 8 7 6 5 i 1 4 3 2 1 2 00 34 2 dx 1m f x 4x where Ax 318 b a n 5 2 3 7 2 4 and x a iAx 9 2 5 X y 9r 8 7 5 3 2 2 0 34 2 2 3 7 2 4 9 2 5 X O 8 y 7 6 5 3 2 2 2 3 2 2 512 3 7 2 4 5

The graph of a function f is given YA 10 1 70 Estimate 1 0 1 f x dx using five subintervals with the following a right endpoints b left endpoints XXX c midpoints

The graph of a function g is shown Estimate a right endpoints b left endpoints YA c midpoints 0 19 x g x dx with six sebintervals using the following 2 1 X