Differentiation Questions and Answers

Given the side length a 13 and the angle A 41 on the triangle below find the lengths of b and c and the measure of angle B Do not round during calculations Round your answers for lengths of b and c to one decimal place Round your answer for angle B to the nearest whole number B Provide your answer below b c B 13 C b C 41 A

A pizza pan is removed at 1 00 PM from an oven whose temperature is fixed at 450 F into a room that is a constant 71 F After 5 minutes the pizza pan is at 300 a At what time is the temperature of the pan 125 F b Determine the time that needs to elapse before the pan is 200 c What do you notice about the temperature as time passes a The temperature of the pan is 125 F at PM

On the graph of f x cot x and the interval for what values of X does Provide your answer below TT 3 TT 2 2 1 1 2 0 0 1 2 3 2 0 TT 2 TT

ow and b w 2 2 2 at the point x y z 2 1 2x if w x y z and y sin z z sin x 0

Suppose that 3x 2y 1 and x tcos 8 Find F and at dy

Use an integration formula to find f O 16 O w a ln 164 cos O 16 r 4 In V x 1 16 4 C 3 4 16 2 C C 16 x dx

Problem 26 Suppose 7 R R be a linear transformation whose standar matrix has the basis 0 1 2 1 0 1 2 1 1 Find an orthonormal basis R use the usual inner product on R THE

Find the partial fraction decomposition of the rational function x 14 x 2x 8 2 x 2 3 2 2 2

f f 3 For the function f x y sin 5x 8y find and dx dy f x f 0

Given that sec x 2 what is sec x

a Write out the form of the partial fraction decomposition of the function as in this example Do not determine the numerical values of the coefficients x6 x2 4 b x x 1 x 8

Problem 6 Sketch a graph of the function f that satisfies the following conditions f is continuous on f 2 1 f 0 0 f 2 1 f 0 0 f x 0 on 0 f x 0 on 0 f 2 0 f 2 0 f x 0 on 2 and 2 00 f x 0 on 2 2 lim f x 2 lim f x 2 8118 218 Solution

3 Use a half angle formula to find the exact value of sin 8 sin 3 8 0 3 X G

On the graph of f x cos x and the interval 2 0 for what value of x does f x achieve a maximum Choose the point in the figure Provide your answer below 24 2TT 1 5 0 5 AAA 3T 2 TT T 2 0 5 1 n 1 1 5 3f1 2 2TT

Find the arclength of x 3y 201 on 2 y 6 to three decimal places

On the graph of f x cos x and the interval 2 4 for what value of x does f x achieve a minimum Choose the point in the figure Provide your answer below 24 1 5 4 0 5 0 0 5 0 1 TI TT 31 2 2TT 5TM2 3TT 71 2 4TT

A cylindrical package to be sent by a postal service can have a maximum combined length and girth perimeter of a cross section of 108 inches Find the dimensions of the package of maximum volume that can be sent The cross section is circular radius length in in

Consider the following function f x 6x 4x Find the limit lim Ar 0 f x Ax f x Ax

If y In e then y 1 True O False

Find the nth Maclaurin polynomial for the function f x sin x n 5 Ps x x 3 1 x 1 5 120 t

a Use the Product Rule to find the derivative of the function Select the correct answer below and fill in the answer box es to complete your choice OA The derivative is x 6x OB The derivative is OC The derivative is OD The c ne derivative is O E The derivative is x6 x6 6x5 x b Multiply the expressions x6x7 Simplify your answer Now take the derivative of the answer from the previous step and simplify the answer from part a Check to make sure that the two results are the same That is using either approach dy dx

In t years the population of a certain city grows from 200 000 to a size P given by P t 200 000 30001 dP a Find the growth rate dt b Find the population after 10 yr c Find the growth rate at t 10 d Explain the meaning of the answer to part c a b The population after 10 yr is Simplify your answer c The growth rate at t 10 is Simplify your answer d What is the meaning of the answer to part c OA The growth rate tells the difference between the rate of growth at the beginning of t 0 and t 10 B The growth rate tells the average growth from time t 0 and t 10 OC The growth rate tells the rate at which the population is growing at time t 10

Find the derivative dx 4 X d 4 5 5 X

Find the linearization L x y of the function f x y 2x 2y 4 at each point a 0 0 b 1 2 a L x y Type an exact answer using as needed

5 Another set of conditions that will prove that a quadrilateral is a rectangle is the following Any parallelogram with at least one right angle is a rectangle Given EFGH is a parallelogram with ZH being a right angle Prove EFGH is a rectangle use of a paragraph proof here is nice E

28 Macmillan Learning a Suppose a researcher runs a t test of the difference of two independent means The samples are taken from populations of unknown variances that are not assumed to be equal The sample sizes and standard deviations are summarized in the table Sample Group Sample size m 22 m 47 df standard deviation 83 8742 32 5 7356 Use software to compute the Welch Satterthwaite approximation of the degrees of freedom df You might find some software manuals useful Give your answer precise to at least three decimal places Attemp

After applying L Hospital s Rule three times we have lim sin x So we still have an indeterminate limit of type x 0 96x find additional derivatives The derivative of sin x with respect to x is The derivative of 96x with respect to x is We will apply L Hospital s Rule for a fourth time To do so we need to

x sin x 16x3 we will do so here applying the rule for a second time To do so we need to find additional derivatives The derivative of x sin x with respect to x is After applying L Hospital s Rule once we have lim x 0 The derivative of 16x3 with respect to x is So we still have an indeterminate limit of type olo 0 We recall that L Hospital s Rule can be applied repeatedly as needed and

Suppose that 11 000 is invested at continuous rater an grows to 81 108 in 22 years Find the interest rate r The interest rate is Round to the nearest percent as needed

Compute the following using a calculator sin 13 Round your answer to two decimal places

Use the quotient to miu 8x 7 x y Choose the correct answer below OA 8 x 8x 7 OC 8x 7 2x x OB 7 2 xX x 7 OD 16 X 8 X X

the derivative of the function x 3 2x 5 5x 1 1 plify your answer

the function using one or more of the differentiation rules 5 14

The function s t 612 91 2 gives the distance from a starting point at time t of a particle moving along a line Find the velocity and acceleration functions Then find the velocity and acceleration at t 0 and t 5 Assume that time is measured in seconds and distance is measured in centimeters Velocity will be in centimeters per second cm sec and acceleration in centimeters per second per second cm soc v t Do not factor a t v 0 cm sec v 5 cm sec a 0 cm sec a 5 cm sec EIIE

For the function f x x 5 find f x f 0 f 1 and f 2 f x Select the correct choice below and fill in any answer boxes within your choice OA f 0 OB f 0 is undefined Select the correct choice below and fill in any answer boxes within your choice OA f 1 OB f 1 is undefined Select the correct choice below and fill in any answer boxes within your choice OA f 2 OB f 2 is undefined

Find the derivative of the function y 2 In 2x 2 3x

Find the derivative f x 4 9x 5

Find i the net area and ii the area of the region bounded by the graph of f and the x axis on the given interval You may find it useful to sketch the region f x 5x 5x on 0 3 1 The net area is Type an exact answer using radicals as needed ii The area of the region bounded by the curve and the x axis is GITE

Use the graph to find the following a the domain off G b the range of f c the x intercepts d the y intercept e intervals on which f is increasing f intervals on which fis decreasing g intervals on which f is constant h the number at which f has a relative minimum i the relative minimum of f 1 What is the relative minimum of f 1 f 2 k The values of x for which f x 3 1 Is f even odd or neither 5 00 Type your answer in interval notation f Over what interval is f decreasing 0 5 Type your answer in interval notation g Over what interval is f constant 0 0 Type your answer in interval notation h What is the number at which f has a relative minimum 1 What is f 2 1 2 21 k What are the x values where f x 3 The leftmost x value where f x 3 is when x What is the rightmost x value where f x 3

4 Find the value of a The dot represents the center of the circle a CHEB L 66 20

Determine the exact values of sin 0 and cos if the terminal side of angle intersects the unit circle in the first quadrant Provide your answer below

1 2 cos 300 2 2 2 cos 210

Graph the function g x log x 3 2 O O O 3 3 1 2 1 3 3 3 6 X

completely factor p x into linear functions zero 3 i p x x 4 6x 3 46x 2 216x 360

of For the function f x y sin 3x 3y find 7 dx f 6 21 sin 3x 3y cos 3x 3y dx f av 11 6 21 sin 3x 3y cos 3x 3y and of dy

f 7 For the function f x y sin 3x 3y find dx of 7 and f dy

Find two values of 0 0 0 2 that satisfy the following equation 2 2 cos 0 0 Simplify your answer Type an exact answer using as needed Use integers or fractions for any numbers in the expression Uso a comm

If x S t 4t 2 dt 0 then f x 24 4 0 3 4x 3 2x X

We want to use the Basic Comparison Test sometimes called the Direct Comparison Test or just the Comparison Test to determine if the series k converges or diverges by comparing it with 1 Ab We can conclude that The first series converges by comparison with the second series The Basic Comparison Test is inconclusive in this situation O The first series diverges by comparison with the second series

formula is a formula for calculating the solutions of a quadratic equation