Application of derivatives Questions and Answers

Find the absolute maximum and minimum of the function f x y 2x 4x y 4y 6 on the closed triangular plate bounded by the lines x 0 y 2 and y 2x in the first quadrant On the given domain the function s absolute maximum is

U suppose the supply function of a certain item is DCX 14 x find the producer Surplus given by S X 4x 2 and the demand function option A 16 3 By 8 16

find the area bounded by the given Y 2x x y 2x 4 option A 31 B 32 34 3

valuate the definite integral 7X 2 dx Topsion A 117 By ing G in 17X 21 64 9 n 7 in 2

differential equation sydy 41 find the general solution for the opton A y 15x C B y c G y 3x 6 TS 8 D y 15x 2 c D 3 8

Determine whether the given functions are inverses Explain wh f x 2x 3 and g x C 2 f and g are inverses because f g x x and g f x f and g are not inverses because fg x f and g are inverses because g f x x f and g are inverses because f g x

Below is the graph of y 3 Translate it to become the graph of y 3 2 4 X

Below is the graph of y X x 1 Translate it to become the graph of y 4 X

find the area of the shaded region option A 070 y x ux 3 3 15 Ex D 65 33 Joofstan

6 Consider the following functions a A r r b f x 3 c g x x d h t 3 e k x 5 x Which are power functions Which are exponential functions

will also be adjacent to a river thus no fencing is required along the river see diagram below Determine cent rectangular corrals The corrals the overall dimensions of the corrals such that the enclosed area is a maximum What is the maximum amount of enclosed area 2 e length of the shorter side of the optimal corral is ength of the longer side of the optimal corral is timal corral contains has an area of Y x X meters long meters long square meters

4 3 3 0 2 2 0 0 y f x 4 3 3 5 3 4 5 3 7 6 graph of f to identify the following intervals or points Interval type solutions should be entered

The diagram below represents the rate of making copies on a copy machine Which table represents this rate A C Copies Seconds 24 Time seconds 6 12 18 Time seconds 1 2 3 Copies 1 2 3 Copies 4 12 B D Time seconds 1 2 3 Time seconds 4 12 Copies 6 12 18 Copies 1 2 32

Determine the area of the given region y cos x y 1 5 1 0 0 5 KI6 6 4 KI3 KIN 2

Consider the following If an answer does not exist enter DNE f x In 2 sin x 0 x 2 Find the interval s on which f is concave up Enter your answer using interval notation DNE Find the interval s on which fis concave down Enter your answer using interval notation X Find the inflection points of f smaller x value larger x value x y x y

A particle moves along a line with a velocity v t t 3 measured in meters per second Find the total distance the particle travels from t 0 seconds to t 5 seconds

r 3 5 0 P x y S 2 1 Using the image above find the y coordinate of point P Give your answer as a decimal rounded to 2 places after the decimal point

The graph of the derivative f of a function f is shown YA 2 y f x A 6 X a On what intervals is f increasing Enter your answer using interval notation On what intervals is f decreasing Enter your answer using interval notation b At what values of x does f have a local maximum or local minimum Enter your answers as a comma separate

Use the given information to sketch the graph of f Domain All real x except x 4 and x 4 f 6 5 f 0 0 f 6 5 f x 0 on 4 and 4 oo f x 0 on 4 4 f x 0 on 4 and 4 0 f x 0 on 0 4 and 4 Vertical asymptotes x 4 and x 4 Horizontal asymptote y 0 Choose the correct graph below OA B Ay 20 to 10 20 OB 1120 10 B 4 Ay 10 20 o OC 84 20 10 A10 20 AY X 57

find the area between the curves x 2 x 1 y 11x y x 12

Match the graph of f with the correct sign chart Choose the correct sign chart below OA f x OC f x 0 3 ND X X OB f x OD f x ND 3 O 3 Af x

Match the graph of f with the correct sign chart O C f x Now use all of the gathered information to choose the corred sign chart below O A f x ND 6 046 6 10 X 6 OB f x OD f x 12 0 6 ND 6

The graph of y f x is shown to the right Identify the x coordinates of the points where f x has a local minimum a OA d g OC b f C e f g h Q Which of the following shows every x coordinate where f x has a local minimum Choose the correct answer below OB a d g OD a c g

Billy needs to construct a cylinder from a single sheet of metal with a perimeter of 186 mm Determine the dimensions of a sheet of metal such that the cylinder created with it is of maximum volume x x y Y

A student claims that the line tangent to the graph of g x In x at x 2 passes through the origin Is she correct Will the line tangent at x 3 pass through the origin Explain Is the student correct in saying that the line tangent to the graph of g x In x at x 2 passes through the origin O Yes No Cannot be determined Will the line tangent at x 3 pass through the origin Explain the line tangent at x 3 pass through the origin because the tangent line is Ag x 10 18 716 H the line passing through 0 0 and 3 f 3 H Q G

Given y 5x 2x find dy dt dy dt when x 1 and Simplify your answer dx dt 3

Next question D x is the price in dollars per unit that consumers are willing to pay for x units of an item and S x is the price in dollars per unit that producers are willing to accept for x units Find a the equilibrium point b the consumer surplus at the equilibrium point and c the producer surplus at the equilibrium point 169 S x x D x a What are the coordinates of the equilibrium point Type an ordered pair

Merryll is a rancher and would like to construct a corral along the north side of a river No fencing is required along the river The west side of the corral is adjacent to the neighbor s property who has agreed to evenly split the cost of the fence along the west side of the corral If Merryll would like to spend no more than 1 275 000 on the project and has determined that the full cost per foot of fencing is 50 per linear foot determine the dimensions of the corral of maximum area given the cost constraint feet Determine the length of the east and west sides of the corral Neighbor s Land feet Y Determine the length of the side of the corral that runs parallel to the river Determine the maximum area of the corral square feet x

Find a function of the form y A sin Bx Cor y A cos Bx C whose graph match shown below 45 44 4 22 40 Leave your answer in exact form if necessary type pi form P

year 1980 1982 1985 1990 price 3 68 5 17 6 12 5 8 Question 8 01 980 through 2005 1995 1998 2000 2003 2004 2005 6 06 6 82 7 76 9 52 10 74 13 84 Graph a scatterplot of the data How many concavities are shown 03 years 1 pts

A landscape architect would like to design a rectangular flower garden with an area of 30 square feet is surrounded by a brick border 6 feet wide on two sides and 5 feet wide on the other two sides see figure What dimensions of the garden minimize the combined area of the garden and borders What is the optimal area 1 y X The length of the shorter side of the optimal garden is Garden The length of the longer side of the optimal garden is The optimal garden has an area of Submit Question feet squared feet long feet long C

13 Find two negative and three positive angles expressed in radians for which the point on the unit circle that corresponds to each angle is 17 Choose the correct angles below O A OB OC O D 14x 3 18x 5 10x 3 10x 3 W 8 2 8 3 3 3 14x 3 2x 2x 8x 18x 5 5 5 5 4x 2x 4x 14x 3 3 3 3 4x 2x 8x 14x 3 3 3 3

Olivia needs to access a point 3 km downstream on the opposite side of a straight 6 km wide river See diagram below Start End She can row at rate of 6 km per hour and run at a rate of 12 km per hour Olivia has a few basic options for this trip She can row straight across the river and then run downriver to her destination She can row the entire distance to her destination She can row to a point x km downriver and then run the rest of the way to her destination How far downstream on the opposite side of the river should Olivia land in order to minimize the amount of time required to arrive at her destination

Suppose that p x y represents the production of a two product firm The company produces x units of the first product at a cost of c each and y units of the second product at a cost of c each The budget constraint B is a constant given by the following formula Use the method of Lagrange multipliers to find the value of in terms of p Py C and c The resulting equation holds for any production function p and is called the Law of Equimarginal Productivity B C X C Y Choose the correct answer below OA Px C Py C Px Py OB O C OD C C Py Px C C C C Px Py W

Rafaela needs to tether two transmission poles using wire and a single anchor between the poles If one pole is 32 feet tall the other pole is 99 feet tall and the poles are 131 feet apart determine the optimal placement of the anchor such that the least amount of wire is used x Anchor

ngelo needs to design a box such that the sum of the box s girth and length does not exceed 678 centimeters iven this constraint determine the dimensions and volume of the box of maximum volume cm cm x x

An open top box is constructed by starting with a square section of material 150 mm by 150 mm removing a small square of material from each corner and folding up each edge See the diagram below Length Determine the dimensions of such a box such that the volume is maximal Width w Height mm mm W mm Determine the maximum volume of such a box w 2x

A landscape architect is designing an 1125 square foot rectangular garden The perimeter of the garden is to be lined with shrubbery on three sides which costs 72 per foot whereas the fourth side of the garden is to be enclosed with fencing at a cost of 53 per foot Determine the dimensions of such a garden with a minimal cost What is the minimum cost Y x y Minimum Cost X Garden Y X

Find the indicated maximum or minimum value of f subject to the given constraint Minimum f x y 9x y 2xy 17x 2y y x 1 The minimum value is

Determine the domain for the function of two variables h x y 6x e Choose the correct answer below O A x y y 0 O B x y y2 6 OC x y y 6 OD x y y 0

Find fx fy and f The symbol is the Greek letter lambda f x y 2 4xy 8x 11y 2

For f x y In x y find f e 10 f e 10

Next question Two variable quantities A and B are found to be related by the equation given below What is the rate of change dA dt at the moment when A 4 and dB dt 5 A B 65 dA when A 4 and dB dt 5 dt Simplify your answer

Find the zeros and x intercepts of f x x 8x 9 O Zeros 1 3i x intercepts 1 O Zeros 3i 1 x intercepts 1 O Zeros ti 3 x intercepts E3 O Zeors 13 11 x intercepts 3 1

Find the extremum of f x y subject to the given constraint and state whether it is a maximum or a minimun f x y 4x y 2xy x y 7 Find the Lagrange function F x y F x y Find the partial derivatives Fx Fy and Fx Fx Fx 11 11 11 There is a value of located at x y Type an integer or a fraction Type an ordered pair using integers or fractions

9 Let S r r is a positive number such that o k k converges absolutely for 2 r Which of the following statement is true a S is empty b S is nonempty and r 1 for all r S c S is nonempty and if for some re S a function f is defined as f x 2 k 0 for IE r r then f x f x for x 0 r d S 0 00 e None of the above statements is true

Find the absolute maximum and minimum values of the function subject to the given constraints k x y x y 4x 4y 0 x 3 y 20 and x y 6 The minimum value of k is Simplify your answer The maximum value of k is Simplify your answer

Let f x 8x 3x 2x 6 Find the following Degree of the f x Leading coefficient 5 3 End behavior Note type infty for o and infty for As f x As a o f x Maximum number of intercepts Maximum number of turning points

If f x y xy find the gradient vector Vf 7 5 Vf 7 5 Use the gradient vector to find the tangent line to the level curve f x y 35 at the point 7

Pictured below are a contour map of f and a curve with equation g x y 8 Estimate the maximum and minimum values of f subject to the constraint that g x y 8 maximum minimum g x y 8 70 60 50 40 30 20 10