Differentiation Questions and Answers

Product Rule f x g x f x g x g x f x dx Quotient Rule f x dx g x No 1 Find a f x b cus x F x g x f x f x g x g x the derivative 5 Sins cus S y 5 In cos sx 5 C

1 Find a f x b cus x F x the derivative 5 Sins cus 5 y 5 In Los 5x 5 sesx 5x 1

A water tank is leaking water from a crack in its base The amount of water in hundreds of gallons remaining in the tank t hours after the crack formed can be modeled by W a decreasi function of time t Which of the following gives a verbal representation of the function W1 the inverse function of W A W is an increasing function of the amount of time after the crack formed W is a decreasing function of the amount of time after the crack formed W is an increasing function of the amount of water in the tank 1 W is a decreasing function of the amount of water in the tank

Which of the following includes the input output pairs 2 4 and 3 8 A B D The arithmetic sequence an 4n The linear function f n 2 4 n 1 The geometric sequence gn 2 n 1 The exponential function h n 2 2 1

The function f is given by f x 27 and the function g is given by g x f For which of the following transformations is the graph of g the image of the graph of f A B D A horizontal translation to the left 3 units A horizontal translation to the right 3 units A vertical translation up unit A vertical translation down unit

The functions f and g are given by f x 2 and g x 2 2 where a 0 Which of the following describes the relationship between the graph of f and the graph of g A B C D The graph of g is a vertical translation of the graph of f by a units The graph of g is a horizontal translation of the graph of f by a units The graph of g is a vertical translation of the graph of f by a units The graph of g is a horizontal translation of the graph of f by a units E

Problem 3 1 point Find a simplified expression for sin 2x 1 cos 2x A tan r B cotx C sec x D CSC C E 1

Problem 5 1 point If f x ln lnx find f e D In 2 E 1 2 B C 2 10 1 A 27 Problem 6 1 point Find the area of the region enclosed by the curves y 360x a y 10x A 6426 B 6480 C 6462 D 6372 E 6444

Problem 29 1 point Which of the below areas are equal to the following limit 2 lim n i 1 3 2 3i n i The area under the graph of 2 x from x 0 to x 3 ii The area under the graph of from x 2 to x 5 iii The area under the graph of 1 x from x 0 to x 3 A only i B only ii C only i and ii D only iii E only ii and iii C F

Problem 8 1 point If the tangent line to y f x at 2 1 passes through the point 1 1 find f 2 and f 2 A 2 and 1 B 2 and 1 C 1 and 1 D 2 and 0 E 1 and 0

Problem 4 1 point Let f x sin x2 Which of the following statements is true A f is invertible on 7 0 1 1 D f is invertible on 0 B f is invertible on 2 0 C f is invertible on E f is invertible on 0 2

Problem A 22 2 5 1 point If f x ln lnx find f e B 12 C 2 1 D In 2 E 1 2 e

Find each limit Use oo or oo when appropriate 6x 6 x 6 A lim f x x 6 f x 2 B lim f x x 6 OA im f x x 6 A Select the correct choice below and if necessary fill in the answer box to complete your choice OA lim f x Simplify your answer x 6 OB The limit does not exist B Select the correct choice below and if necessary fill in the answer box to complete your choice Simplify your answer C lim f x x 6 BOXER OA lim f x Simplify your answer x 6 OB The limit does not exist s no exist C Select the correct choice below and if necessary fill in the answer box to complete your choice OB The limit does not exist

A function f x is given below Express the relationship between f x and f x in words and write a differential equation that f satisfies x 6 CTXH Express the relationship between f x and f x in words Choose the correct answer below OA The derivative of f x is 6 times f x OB The derivative of f x is 1 times e raised to the x 1 power OC The derivative of f x is the same as f x OD The derivative of f x is 1 times f x f x satisfies the differential equation y Type an expression using y as the variable

Write an absolute value equation for the graph shown to the right y Simplify your answer 12 10 8

Assume f x is continuous on 0 00 Use the given information to sketch the graph off 2 4 7 4 1 0 1 x 1 x X 1 x 4 5 ND 0 0 0 0 ND 2 Choose the correct graph below OA 6 1 OB 0 6 0 5 LICE O C OD

A candy box is made from a piece of cardboard that measures 13 by 8 inches Squares of equal size will be cut out of each corner The sides will then be folded up to form a rectangular box What size square should be cut from each corner to obtain maximum volume EXXE A square with a side of length inches should be cut away from each corner to obtain the maximum volume Round to the nearest hundredth as needed

Find f x f x and f x of f x x 3x 2 8x 4

Suppose that the company sports stylz Inc determines that the cost in dollars from the sale of X sunglasses is given by C X 0 05x 50x Find the average rate of change from selling 300 Phones to and X 301 301 Phones That is X 300 Average rate of change F xa f x X X

T X x 10 x 11 ou

Sketch the graph of the continuous function f that satisfies all of the stated conditions 9 pts brodo hav 201 f 0 2 f 1 0 2 2 f 0 0 f 2 0 bution of for en f x 0 on 0 and 2 00 f x 0 on 0 2 baralimos w f 1 0 6 5 f x 0 on 1 0 f x 0 on 0 1 whoito 120 TOIDUOTOS TOY otrd bouquio pr 18 4 cat 3 A 5 3 2 1 ON 411 2 23 3 4 5 6 2 A 6 305

Consider the demand function given by D x 240 20x where D x is the quantity of coffee mugs demanded at a price of x dollars per mug 5 pts a Find the elasticity E x D x D x b At a price of 2 should the company raise or lower the price to increase revenue Explain your answer

the function f x x 3x 1 a Make a sign diagram for the first derivative b Find any relative maximum or relative minimum points Write the points as an ordered pair x y Relative Max Relative Min c Make a sign diagram for the second derivative d Find the inflection point s if there are any Write the points as an ordered pair x Inflection Point s e Graph the function Clearly show any relative maximum relative minimum and inflection points and at least 5 additional points on the graph 10 10 8 6 4 2 thon 8 6 4 2 2 4 6 8 10 2 4 6 8 10 X y toupe no

3 Find the derivative of the given functions 24 pts a f x 3x 2x 1 Expur 0000 and iw enixolen bno luthsbnow s In 4x

Given the F X axty find the equation of Point 116 tangent line of F X at the Steps M Slope F X lim f xth f x hs0 Find the slope you will get a Find the equation y y m x x the

A patient receives an injection of 9 5 milligrams of a drug and the amount remaining in the bloodstream A t 9 5e 0 04t Find the instantaneous rate of change in milligrams per hour of this amount under the following condition Round your answers to two decimal places a immediately after the injection time t 0 milligrams per hour b after 5 hours milligrams per hour

The number of traffic accidents per year in a city of Population is predicted to be A 0 002 p312 If the population is growing by 500 people per year find the rate at which traffic accidents will be rising when the population is P 40 000

suppose that the daily cost in dollars of producing X radius radius at a current production level of 40 racios is C X 0 002x 0 1X 42x 300 C 40 a what is the current daily cost b what is the marginal cust when X 40 C use marginal cost estimate the daily cost of increasing Production to 41 radius daily C 41 C 40 C 40 C 40

F X 3 X4 X Find second derivative

F x x 3x 2x 8x 4 Find first second third derivative

Find the derivative of G X 3x 4x 5 using Chain ful

7 6 points Use the following information to produce a graph of the function f over the closed interval 2 5 The graph of f is continuous The graph starts at the point 2 3 The derivative of f is the function graphed here 1 2 2 1 1 1 3 2 1 1 2 Graph of f x 1 2 Graph of f x 3 3 4 4 5 5

1 Determine the average rate of change of y in the function y x 7x 10x 16 over the interval 1 5 5 marks 2 The distance travelled by an object is modelled by the function d t 6t 60t 151 where di conferents the distance natres and converents the time in raconde Estimata

4 THE SUBSTITUTION RULE If u g x is a differentiable function whose range is an interval I and f is continuous on I then f f g x g x dx f f u du EXAMPLE I Find x cos x 2 dx u x 2 so du 4x dx x dx du 4 x cos x 2 dx cos u du cos u du sin u C sin x4 2 C 8 x x 5 dx 10 3 2 4 dt C

1 2 I zx x n

13 An apple orchard now has 30 trees per acre and the average yield is 400 apples per tree so the crop is 12 000 apples per acre For each additional tree planted per acre the average yield per tree is reduced by 10 apples How many trees per acre will give the largest crop of apples HINT Let X be the number of additional trees per acre Then there are 30 X trees per acre and the yield per tree is 400 10X This is enough to determine the crop per acre which is what you want to maximize 007 3 LOST A 30 B 35 C 40 D 45 atute 2 2122 pouth seanq 734 Shton 30th ya

11 A plane flying horizontally at an altitude of 9 mi and a speed of 470 mi h passes directly over a radar station Find the rate at which the distance from the plane to the station is increasing when it is 10 mi away from the station

car moves along the x axis with position from the starting point given by some function S t If the acceleration a t S t is positive then the SPEED is A Is increasing B Is decreasing C Can t tell without more information i e may be increasing or decreasing

7 If g x tan sec x then g x 02 192 A 3 tan secx B 3 tan secx tan x C 3 tan sec x sec I D 3 tan secx sec sec x sec x tan x E 3 tan secx sec sec x sec x

6 If f x 2x43x 9 then f x old xare pul daizs fon 290 A 2x 3x B 8x 6x C 8x 6x 9 D 5

A wooden artifact from an ancient tomb contains 45 percent of the carbon 14 that is present in living trees How long ago to the nearest year was the artifact made The half life of carbon 14 is 5730 years years Question Help Video Message instructor Submit Question

X 1 3 4 7 g x 2 1 7 9 C 2 g x 3 8 5 2 Let and gbe inverse functions wheref g 1 The table above gives selected values for g and its derivative g Find f 7 1 8 D

Part III No Calculators Allowed 11 Consider the curve given by 2y2 3xy 2x 12 dy dx a Show that 6x 3y 3x 4y b Is there a horizontal tangent to the curve where x 1 Justify your answer d y c Find at the point 0 6 dr

8 x9 zx x a 0

3 In the development of a new drug a certain medication concentration in the bloodstream C t measured in miligrams per mililiter is modeled by the function C t 1 50 t 4t 8 where t is the time in hours since the drug was administered How much time does it take for the medication concentration to reach its maximum

a 3 points Sketch the graph of f x 2 e by transforming the graph of y e Include any relevant asymptote on your graph 30 y 0 OR X 0

Find the particular solution of the differential equation dy 5 dx 1 x 1 x y 0 9

Refer to the graph to the right and follow the directions in parts a through f a Is a 1 or is 0 a 1 b Give the domain and range of f and the equation of the asymptote c Sketch the graph of g x a d Give the domain and range of g and the equation of the asymptote e Sketch the graph of h x a f Give the domain and range of h and the equation of the asymptote a Is a 1 or is 0 a 1 Choose the correct answer below 1 a y f x a 1 a 0 1 O OA For the function f x a 0 a 1 because as a gets closer to 0 the graph becomes steeper moving to the left of the y axis OB For the function f x a a 1 because as a gets closer to 0 the graph becomes steeper moving to the left of the y axis OC For the function f x ax a 1 because as a becomes larger the graph becomes steeper moving to the right of the y axis OD For the function f x a 0 a 1 because as a becomes larger the graph becomes steeper moving to the right of the y axis b Give the domain and range of f and the equation of the asymptote The domain of f is

10 O A 10 2 8 10 10 2 8 5 8 1 8 10 10 Q B 10 1 8 3 0 10 10 S O C 10 10 2 5 0 3 10 O D 10 10 2 5 0 3

16 0 16 Type an integer or a simplifed fraction