Calculus Questions

Suppose a company s revenue function is given by R q q 390g and its cost function is given by C q 350 12g where g is hundreds of units sold produced while R q and C q are in total dollars of revenue and cost respectively A Find a simplified expression for the marginal profit function Be sure to use the proper variable in your answer MP q B How many items in hundreds need to be sold to maximize profits Answer sold Bound to two decimal hundred units must be

What is the value of g 0 if y g x has a graph given by 2 1 10 OA O OB 4 O C 3 O D 1 OE 2 432 423 2 4 5 y g x 2 Click here for long description

Assume the function f is defined as f x 5cosx 4sinx Use the corresponding derivative rules to find the second derivative f x and evaluate that second derivative at x 1 7 Be sure your calculator is in radians mode retain at least eight decimal places in your calculations Use the central difference formula for the second derivative to estimate f x at x 1 7 for four different values of the mesh sizes h 0 2 h 0 1 h 0 05 h 0 025 Answer the following two questions Does the error in the central difference approximation drop as the mesh size h is reduced As the mesh size h is reduced by a factor of 2 by what factor is the error in the central difference approximation reduced

Assume the function f is defined as f x 5 cos x 4 sin x Use the corresponding derivative rules to find the second derivative f x and evaluate that second derivative at x 1 7 Be sure your calculator is in radians mode Note To avoid round off error retain at least eight decimal places in your calculations Use the central difference formula for the second derivative to estimate f x at x 1 7 for four different values of the mesh sizes oh 0 2 h 0 1 o h 0 05 oh 0 025 O

Find the derivative of the given function dx 3 y 8 cos x

Find the derivative of the given function 3 y 5 log x dy 11 Type an exact answer

Find the derivative of the given function V 20 tan s 1 dV ds Type an exact answer using radicals as needed

Find the differential of the function y The differential is dx 12eX X 7

Find the derivative of the given function 2 y 3 sin x sec x Choose the correct answer below O A O B O C O D dy 6 sec x 2x cos x tan x sin x dx dy 2 2 6 sec x x sin x tan x cos x dx dy 2 2 3 sec x 2x sin x tan x cos x dx dy dx 2 3 sec x sinx 2x tan x 1

Evaluate the limit if it exists Use L Hospital s rule if appropriate lim X 0 lim x 0 In cos 4x 3x In cos 4x 3x Simplify your answer

Find the derivative of the given function r sin 30 dr Type an exact answer using a pooded

Find the derivative of the given function y 3 sin 6x dy dx 0

Find the derivative of the function by using the product rule Then multiply out the function and find the derivative by treating the function as a polynomial Compare the results y 5x 4 2 x ax ax d d 2 x 5x 4 5x 4 dx 2 x dx 5x 4 2 d d 2 x 2 x 5x 4 dx dx D 5x 4 2 The derivative using the product rule is y 14 10x Multiplying the function out results in the polynomial 5x 14x 8 Simplify your answer Do not factor The derivative of the polynomial is y 10x 14 Compare the results Choose the correct answer below OA The derivatives found by both methods are the same OB The derivatives found by both methods are negative inverses

Find the derivative of the function Do not find the product before finding the derivative y 4x 3x5 5x Choose the correct answer below OA y 4x 15x4 10x 3x5 5x 8x OB y 5x 4x 3x5 5x 15x4 10x OC y 3x5 5x 15x 10x 8x 4x OD y 4x 3x5 5x 15x4 10x 8x

Find all the higher derivatives of the following function y 6x 4x dy dx d y 2 dx 11 18x 8x

Find the derivative dy y 4 24x 2

Find the derivative of the given function y 3sin 3x4 5 dx

Find the second derivative f x 5x 3

Find the derivative of the function by using the quotient rule 4x 1 x 9 2 y y

Evaluate the derivatives of the given function for the given value of x Check your results using the derivative evaluation feature of a graphing cal y 4x 5x 4 3 x 5x x 0 0 I C a fraction

Find the derivative dy 10x 7 3 2 y

Differentiate the function f x 2x 3 with respect to x

If the following function is continuous what is the value of a b 3t 2t 1 h t A 2 OB 4 OC 0 OD 1 E 3 ift 0 a cos t b if 0 t 3 ift 2 4 sin t

If g x is continuous for all real numbers and g 3 1 g 4 2 which of the following are necessarily true I g x 1 at least once II lim g x g 3 5 x 3 5 III lim g x lim g x x 3 x 3 OA I only OB II only C I and II only D I II and III E None of these

lim cos x x 0 X O A O OB 1 2 O C 1 OD 2 OE Does not exist

2 x x lim xoo 2x 4x4 3 O OB 1 O C 0 O D 2 O E A 4 00 c

OA OB O C 38 OD OE 7 2 4 a 2 3

Let f be defined as follows where a 0 x x 2ax a xia 5 ifx a Which of the following are true about f I lim f x exists x a II f a exists III f x is continuous at x a A None ifx a OB I II and III O C I only OD II only OE I and II only

6 8 6 9 6 99 7 01 7 1 7 2 g x 9 44 10 21 10 92 11 08 11 31 12 56 X It would appear that lim g x x 7 OA 11 OB Does not exist OC X 4 O D O E 7

Find the general antiderivative of f x 2x 1 6 3 x2 x

lim X 5x x7 et O A OB 5 3 OC 0 O D 1 OE 1

lim 818 12 2 3x 4 O A 0 O O B 2 OC 2 D 3 OE 3

lim 49 h 7 h 0 h O A 14 B 7 OC 0 OD 14 O E T

The function f x x bl intersects g x 2 x 4 at exactly one point The value of bis A 4 OB 5 C 5 D 4 E Any real number

If f x 3 then f x 3 x 3 OA 3 X OB No inverse exists O c 3 2 O D O E 3 x 3

Which of the following is always true of odd functions 1 f x f x II f xl is even III If x is even OA All of these are true B None of these are true OC I only D II only E I and III only

If g x 3x 1 and f x 9 x then which of the following will have a domain of 3 1 3 U 1 3 3 O A g f x OB f g x O C g f x OD f g x OE g f x

Which of the following is an even function NAVA Graph I Graph III Graph II Click here for long description A I only B II only C Ill only D I and III only

You would probably use calculus to determine the area for which of the following shapes OA C OB O C A D E

You may not use your calculator for this question In which of the following is ya function of x I y 9 x II y x III y x A I only B II only C III only OD I and III only E I II and III

he domain of f x x 6x 7 O A 1 7 B 1 7 O c x 1 U x 7 D 1 U 7 E x 1 or x 7 IS

Write the arithmetic sequence 17 23 29 35 in the standard form an

Online Retail Sales Since the inception of the Web online commerce has enjoyed phenomenal growth But growth led by such major sectors as books tickets and office supplies expected to slow in the coming years The projected growth of online retail sales is given by R t 12 24e 0 136t 0 t 4 where t is measured in years with t 0 corresponding to 2007 and R t is measured in billions of dollars per year Online retail sales in 2007 were 116 billion a Find an expression for online retail sales in year t s t b If the projection held true what were online retail sales in 2011 Round your answer to the nearest integer Enter your answer in billions of dollars billion

Example 6 If z f x y has continuous second order partial derivatives and x r cos a y r sin 0 show that 8 x 8 8 dy r 10 z r 20 1 z r r

se Stoke s Theorem to calculate the circulation of the field F around the curve C in the indicated direction F 3y y z where C is the counterclockwise path around the ellipse 4 36 1 in the xy plane

1 point Use Stokes Theorem to find the circulation of F 4y 5zj 6xk around the triangle obtained by tracing out the 4 0 0 to 4 0 6 to 4 3 6 back to 4 0 0 F dr Circulation

Solve the IVP using Laplace Transform y y 8 t 2km y 0 0 y 0

The equation r t 3 cos t i 3 sin t j t k is the position of a particle in space at time t Find the particle s velocity and acceleration vectors Then write the particle s velocity at t 0 as a product of its speed and direction The velocity vector is v t i j k The acceleration vector is a t Oi j k The velocity vector at t 0 written as a product of the speed and direction is v 0 0 0 0j k Type exact answers using radicals as needed

Using the function f x 5x 2x 3x x 8 and Descartes Rule of Signs which are possible combination positive negative and imaginary zeros Select all that apply 2 positive I negative 2 complex 1 1 positive 1 negative 2 complex 2 positive 1 negative 1 complex 3 positive 1 negative 0 complex 1 positive 3 negative 0 complex

Find a particular solution to the following differential equation Do NOT solve the coefficients y 4 4y 3 7y 6y 2y et 4t sint