Application of derivatives Questions and Answers

Each expression simplifies to a constant a single trigonometric function or a power of a single trigonometric function Use fundamental identities to simplify each expression 1 sin x tan x cos x 2 sec x cos x 3 tan x cos x 4 sec x 1 5 tan x 2 sin x csc x CA 9 G

Evaluate the following expressions Your answer must be in radians written as multiple of 7 Note that is already provided in the answer so you just have to write the appropriate multiple E g if 2 is the answer you should write 1 2 Do not use decimals Your answer should be a fraction or an integer a tan 1 b tan 3 c tan 3

Evaluate the following expressions Your answer must be an angle T 2 0 in radians Give exact answers do not use decimal answers You can enter the number 7 as pi E g if the answer is 2 you could enter pi 2 or 1 2 pi sin 1 sin cos 1 os 1

oblem 3 Evaluate the following integral TT 3 6 cot x In sin x dx In 4 In 3 2 2 Problem 3 In 4 In 3 2 Just Save Problem 3 Enter your ansv as in these exa Submit Problem 3 for Grading Attempt 1 Attempt 2 At

Fill in the blank to complete the trigonometric identity Simplify your answer complete COS

Use an identity to find the exact value of each expression Note You are not allowed to use decimals in your answer sin 2 T 3 3 COS os 7 6 tan TT TT F BI A

Find the sum 5 i 0 41

In the triangle below x 8 Use the Law of Cosines to solve the triangle Round your answers to two decimal places B 12 cm O O O B 18 cm x cm C

A stone was dropped off a cliff and hit the ground with a speed of 80 ft s What is the height of the cliff Use 32 ft s for the acceleration due to gravity ft

5 7C5

1 2 3 6 700 6 2 W 4 70 3 1 5 g x 2 1700 J 6 3 N 0 2 The functions and g have continuous second derivatives The table above gives values of the function their derivatives at selected values of x a Let k x f g x Write an equation for the line tangent to the graph of k at x 3

2 2 20 Points Find the following antiderivatives ln x

Find the value of each trigonometric ratio 33 sin A A Z 36 35 cos Z B 15 C 39 28 35 Y 21 X 34 sin Z Z 24 25 Y 7 X Z 36 cos X 36 45 1 25 Y 27 X

Find the value of each trigonometric ratio 62 cos C 32 C B 64 tan Z Z 27 Y 40 24 45 36 A X 63 tan Z X 18 65 sin C C 14 B Y 30 24 50 48 Z A

Convert each radian measure into degrees 5 18 1 Convert each degree measure into radians 3 45 2 10T 3 4 300

A cell receives nutrients through its surface and its surface area is proportional to the two thirds power of its weight Therefore if w t is the cell s weight at time t then w t satisfies w aw2 3 where a is a positive constant t Solve this differential equation with the initial condition w 0 7 initial weight 7 units w t 3 at 7 X

Use the Laws of Logarithms to combine the expression 7 log x 8 log x 1

Find the function of the form y log x whose graph is given y II 1 0 0 5 0 5 1 0 2 3 1 3

Use the Laws of Logarithms to expand the expression log V 7x y HODINY FR

Use the elimination method to solve the following initial value problem dR dt dJ dt J R R 0 1 J 0 0

Find the sum 7 1 0

A weight is oscillating on the end of a spring see figure The position of the weight relative to the point of equilibrium is given by y cos 8t 3 sin 8t where y is the displacement in meters and it is the time in seconds Find the times at which the weight is at the point of equilibrium y 0 for 0 sts 1 Round your answers to two decimal p Enter your answers as a comma separated list t H Equilibrium 4x4

The length of each of the two equal sides of an isosceles triangle is 10 meters see figure The angle between the two sides is 0 10 m a Write the area of the triangle as a function of 0 2 A 0 2 0 10 m b Write the area of the triangle as a function of and determine the value of 0 such that the area is a maximum A 0 radians

Rewrite the expression in terms of the first power of the cosine Use a graphing utility to graph both expressions to verify that both forms are the same 3cos 2x

Use the product to sum formulas to write the product as a sum or difference 2 sin 20 sin 50

Use inverse functions where necessary to solve the equation Use n as an integer constant Enter your answers as a comma separated list 2 cos x 13 sin x 8

Use the properties of logarithms to verify the equation In 18 2 In 3 In 2 In 18 In 32 2 In 32 2 2 In 3 In 2 X In 3 X

The relationship between the number of decibels and the intensity of a sound S in watts per square meter is given by 10910 10 12 B 10 log10 a Use the properties of logarithms to write the formula in a simpler form B b Use a graphing utility to complete the table Verify your answers algebraically 10 4 10 6 10 8 S n X DO so Po 100 10 10 n 10 12 an 10 14

Use the properties of logarithms to expand the expression as a sum difference and or constant multiple of logarithms Assume all variables are positive

Rewrite the logarithm as a ratio of common logarithms and natural logarithms 9 lage 0 10 a common logarithms k 9 log 10 log a b natural logarithms

Rewrite the expression in terms of In 2 and In 3 In 108

Consider the function f x x e6 f x has two inflection points at x C and x D with C D 3 6 where Cis 3 and Dis 3 6 3 Add Work X X retry this question below Finally for each of the following intervals tell whether f x is concave up type in CU or concave down type in CD C C D D

Complete the table to determine the balance A for 10 000 invested at rate r 2 for t years compounded continuously Round your answers to two decimal places 1 t 1 10 20 30 40 50

Find the x and y intercepts of the graph of the circle If an answer does not exist enter DNE x 10x y 2y 19 0 STEP 1 Consider the equation for the circle x2 10x y2 2y 19 0 Let y 0 Give the resulting equation below 10x 19 0 STEP 2 Solve your equation from Step 1 for x If one or more solutions exist these values are the x coordinates of the x intercepts of the graph of the circle Enter your answers as a comma separated list x x STEP 3 Consider once again the equation for the circle x 10x y 2y 190 Now let x 0 Give the resulting equation below 2y 19 0 STEP 4 Solve your equation from Step 3 for y If one or more solutions exist these values are the y coordinates of the y intercepts of the graph of the circle Enter your answers as a comma separated list y x STEP 5 Complete the problem by listing the x and y intercepts x intercepts x y y intercept x y x y x x smaller x value larger x value

1 If x 0 and y 0 then 2x y al X

Graph the function not by plotting points but by starting from the graph of y e in the figure below 10 range O 10 O y 10 10 y 10 10 y 2 10 10 X 10 O 10 State the domain and range Enter your answers using interval notation domain y 10 10 10 10

Use transformations of the graph of y 1 x to graph the rational function as in Example 2 domain 10 range 10 5 5 10 X 10 X 10 State the domain and range Enter your answers using interval notation 10 5 5 2

150 180 210 120 240 90 270 60 30 2 3 4 300 330 0

Solve the triangle with the given angles and sides 6 points B 52 C 28 a 43 c

Find the area of the parallelogram whose vertices are given below A 1 4 B 4 0 C 11 3 and D 6 7

Find the volume of the parallelepiped box determined by u v and w u 4i 2j V 3i j 3k w i 3k

0 Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of 0 where 2 3 3 81 9x x 3 cos 0 0 1 1 Then find sin 0 and

The price of a home is 180 000 The mortgage on the hous a 30 year fixed rate mortgage at 4 Use this information to fin the amount of the monthly payment Round your answer to the nearest cent Use the formula below to solve the problem P 1 1 xt PMT 859 35 954 83 012 54

18 Estimate to the hundredth the area from 1 to 5 under the graph of f f x approximating rectangles and right endpoints 2 x using four

The population of a colony of mosquitoes obeys the law of uninhibited growth If there are 1000 mosquitoes initially and there are 1200 after 1 day what is a The growth rate Round to three decimal places b The population of mosquitoes after 6 days Round to the nearest whole number c When will the population double Round to the nearest whole number

Given that 2 is a zero find the remaining real zeros Show work WILL be required on the exam No approximate answers will be accepted 3 x 4x 3x 2 Solution 1 Solution 2

5 10 5 5 5 10 For which values in the domain is this graph increasing Give your answer in interval notation Assume the graph extends to finity in both directions Submit

Find the solution to the equation given below Exact values only Try again 4x 5 x 0 4024

What is the conjugate of b if a 3 4i and b 2 3i

Question What is a b if a 3 4i and b 2 3i