Differentiation Questions and Answers

Evaluate the iterated integral rcsc 0 dr d0 JJres R 3 0 rcsc 0 dr d0 Simplify your answer

Draw the solid region whose volume is given by the following double integral Then find the volume of the solid I 01 A 9 dydx V 3 Find the volume of the solid OB OC

Evaluate the following integral x 4 1 x 2 TIT 000 x 4 1 2 ST 000 sin xx sin y sin 4z dy dx dz sin xx sin y sin 4z dy dx dz

Use graphical approximation methods to find the point s of intersection of f x and g x f x In x g x 7x The point s of intersection of the graphs of f x and g x is are Type an ordered pair Type integers or decimals rounded to two decimal places as needed Use a comma to sepa answers as needed

Use a graphing utility to find the point s of intersection of f x and g x to two decimal places Note that there are three points of intersection and that is greater than x for large values of x f x ex 5 g x x The intersection points are Type an ordered pair Use a comma to separate answers as needed Round to the nearest hundredth as needed

It x and find the equation of the line tangent to the graph of f at the indicated value of x Find the value s of x where the tangent line is horizontal f x In 12 9x 2x x 1 f x The equation of the line tangent to the graph of f at x 1 is y Select the correct choice below and fill in any answer boxes in your choice OA The value s of x where the tangent line is horizontal is are Simplify your answer Use a comma to separate answers as needed B The tangent line is never horizontal

Use the price demand equation below to find E p the elasticity of demand x f p 7200 4p

A company determined that the marginal cost C x of producing the xth unit of a product is given by C x x 2x Find the total cost function C assuming that C x is in dollars and that fixed costs are 8000 C x

Calculate 4 62 Give your answer in a bi form

An oil tanker aground on a reef is forming a circular oil slick about 0 2 foot thick To in cubic feet per minute at which the oil is leaking from dV estimate the rate dt the tanker it was found that the radius of the slick was increasing at 0 42 foot per dR dV minute 0 42 when the radius R was 400 feet Find dt C dt Y using 3 14 dV ft min dt Type an integer or a decimal Round to the nearest whole number as needed 2 A xR V 0 2A Tanker R

A box with an open top has a square base and four sides of equal height The volume of the box is 200 ft The height is 3 ft greater than both the length and the width If the surface area is 185 ft2 what are the dimensions of the box The width and length of the box is ft CHE x X

A concert promoter sells tickets and has a marginal profit function given below where P x is in dollars per ticket Thi means that the rate of change of total profit with respect to the number of tickets sold x is P x Find the total profit from the sale of the first 170 tickets disregarding any fixed costs P x 4x 1152 The total profit is coded Y

1 5 points For each statement circle TRUE if the statement is always true Otherwise circle FALSE You do not need to justify your answer a 1 point A point p is an inflection point if f p 0 or if f p is undefined True b 1 point For a differentiable function defined on a closed interval a b the global maxi mum is either at a local maximum or at an endpoint True e 1 point If a function is increasing it has to be concave up C 1 point If p is a point so that f p 0 and p 0 then p is a local minimum True False True False d 1 point If f x is a continuous function with f 1 0 and f 1 4 then based on the Mean Value Theorem f 0 2 True False False False

6 dx 1 x 2 I

Find the partial sum without using a graphing utility 100 40 n n n 1 n 41

Write the first five terms of the arithmetic sequence Use the table feature of a graphing utility to verify your results a 9 d 1 12 a1 32 a3 34 35

Write the first five terms of the sequence Assume n begins with 1 an 3n 1 81 10 1 a3 34 as Determine whether or not the sequence is arithmetic If it is find the common difference d If the sequence is not arithmetic enter Yes the sequence is arithmetic O No the sequence is not arithmetic d

Determine whether the sequence is arithmetic If so find the common difference d If the sequence is not arithmetic enter NONE 6 10 14 18 22 O Yes the sequence is arithmetic O No the sequence is not arithmetic

12 16 x 7 1 hint Ax is the same as h A 1 2m C 1 1 2r Inr C f x lim A 40 A f x lim 1 3cos o 7 2x which of the following will calculate the derivative of f x 2r Ar 2x x Ax 1x 1 Ar 0 2 0 3sin 0 what is f 0 f 0 1 3 cos 0 o 2 x 4r 2x x x 1 x 1 Ar B f 0 B 0 1 2ln D S 1 2m D f 0 2x Ar 2x B f x Ar 1 x 1 2 x 4x 2x D f x x 4x 1x 1 1 3 0 sin 0 o Ar 1 3 0 cos 0 Ar

8 Consider the function h z 3 x 4 x 1 x 1 a Identify the degree of h b How many turning points does h have c Find the y intercept of h d State the zeros of h and their multiplicity e Find one other ordered pair on the graph of h that is neither an z nor a y intercept

THE CHAIN RULE If g is differentiable at x and f is differentiable at g x then the composite function F f g defined by F x f g x is differentiable at x and F is given by the product F x f g x g x In Leibniz notation if y f u and u g x are both differentiable functions ther dy dy du dx du dx Y Y dy Note addition formula to execute next assignment X U 1 x 10 d dx dy du U 1 x du du dy du 28 y ax a In a dx love 10 1 x 2 30 G y e e u e e u y 2X 5

THE CHAIN RULE If g is differentiable at x and f is differentiable at g x then the composite function F f g defined by F x f g x is differentiable at x and F is given by the product O F x f g x g x In Leibniz notation if y f u and u g x are both differentiable functions then dy dy du dx du dx Note addition formula to execute next assignment 0 dx ax a In a 18 h t 1 t 1 4 20 v x 1 x 2

If possible approximate a root of the following functions up to 1 decimal place in the given intervals For this exercise you are allowed to use a calculator 1 Function f x x 2x 5 g x sin x x 1 h x x53x 1 j x x x 12 k x x x 2x x 1 m x cos x x n x x55x 4x 1 p x tan x x 1 q x x 4x 3x 2 Interval 0 4 0 2 0 1 0 30 1 2 0 1 1 2 0 4 2 1

2 If possible approximate a root of the following functions up to 1 decimal place in the given intervals For this exercise you are allowed to use a calculator 1 Function f x x 2x 5 g x sin x x 1 Interval 0 4 0 2

3 Sketch the following sets of reals a The set A represented by the union of intervals 1 2 and 4 6 b the set B represented by the intersection of intervals 3 1 and 2 5 c the set C represented by the union of intervals 3 2 and 4 d the set D represented by the intersection of intervals 1 and 0 5 e the set E represented by the union of the interval 1 4 and the set of rational numbers in the open interval 4 6

x sinx if x 0 h Investigate the continuity of the function f x 0 if x 0 x 0 at

1 Solve the following exercises about continuity Do not forget to check the right limit as well as the left limit when necessary a Determine if the function f x 2x 3x 1 is continuous at x 2

For f x 9x 9 and g x x find the following composite functions and state the domain of each a fog b gof c fof d gog answers as needed B The domain of fog is all real numbers www b gof x 81x 162x 81 Simplify your answer Select the correct choice below and if necessary fill in the answer box within your choice OA The domain of g of is x Type an inequality Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed B The domain of g of is all real numbers c fof x 81x 90 Simplify your answer Select the correct choice below and if necessary fill in the answer box within your choice OA The domain off of is x Type an inequality Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed OB The domain of f of is all real numbers

Given f x 7 x and g x 6x find the following expressions a fog 4 b gof 2 c fof 1 d gog 0 www a fog 4 14 6 Type an exact answer using radicals as needed Simplify your answer b gof 2 42 2 Type an exact answer using radicals as needed Simplify your answer c f of 1 Type an exact answer using radicals as needed Simplify your answer

h X 2 x 3 1 2

0x45x 4 2 C

a x 2x 11x 12 0

Choose the correct pair of functions OA f x x g x O B f x x g x 3x 8 X 8 3 OC X 8 3 f x g x x O D f x 3x 8 g x x

Between 8 30 and 9 30 a m cars drive through the Lincoln Tunnel at a rate of 12 cars per hour 0 2 per minute The formula below can be used to determine the probability P that a car will arrive within t minutes of 8 30 a m Determine the number of minutes needed for the probability to be 0 6 P 3 9 000 165 P t 1 e 0 2 OR ad of bespieg

entify the property of a gas that is measured in each of the llowing measurements lo Part A 425 K pressure amount of gas O volume temperature Submit Request Answer Part B 010 1 00 atm temperature volume pressure amount of gas

Let co ci x a c x a cn a be the Taylor series of the function f x x sin x For a 0 determine the value of c3

1 point The function f x is approximated near x 0 by the second degree Taylor polynomia P x 3 3x 8x Give values f 0 0 f 0

If y k 1 x 3 then y 0 k 0

Write an expression for the apparent nth term an of the sequence Assume that n begins with 1 1 16 an 11 1 1 F 2 4 8 I

A 2 30 L container is filled with methane gas CH4 at a pressure of 3000 mmHg and a temperature of 25 C Part A How many grams of methane are in the container VAE m Xb b x xx x IXI X 10n X pay 15 66 1023 You have already submitted this answer Enter a new answer Jest Incagain g

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The Narso ring is a circular test track for cars It has a circumference of 12 5km Cars travel around the track at a constant speed of 100km h A car starts at the easternmost poijt of the ring and drives for 22 5 minutes at this speed what is the distance in km does the car travel

Solve the equation and express the solution in exact form In 4x 3 In 9 In x 3 O A OB O C 0 15 15 19 4 3 2 OD

The logarithm of a number raised to a power is the same as the power times the logarithm of the number So log5 2512 12

Use the Change of Base Formula and a calculator to evaluate the logarithm correct to six decimal places Use either natural or common logarithms logg 593

The table below shows values for four differentiable functions Suppose we know the following things b x t x t x p x p x c x c x b x 0 c x 0 b x p x 2 4 0 t x 0 1 What is and What is 1 2 3 4 2 4 1 3 0 4 3 1 3 2 32 S p x dx 4 b x 4 dx What is the equation of the tangent line to the curve y c x when x 0

If f f x dx 8 and Submit Answer If 20 Points f x dx DETAILS f x dx 33 and f x dx 5 9 find 9 6 F X ald du 13 SCALCET8 5 2 049 f x dx 9

Find the number of moles of H in 1660 mL H gas n VAEO 2 mol

At what point is the function f x 16 x not differentiable

ST Find 10 2 if sin x