Application of derivatives Questions and Answers

14 Solve for x Round to the nearest tenth if necessary 10 30 x Not drawn to scale

Prove the following at art artar arn 1 3 using mathematical induction 1 1

Which of the following can 01 sin e 01 sin e sin e cos 8 0 1 sine be simplified to

What are the fourth roots of 2 3 2i Enter your answer by filling in the boxes Enter the roots in order of increasing angle measure cis cis cis

Suppose a conic section passes through the point 2 6 If the conic section has focus 0 3 directrix 4 then it is a ellipse parabola hyperbola O cannot be determined

You owe 302 87 on your credit card Your average daily balance is 112 98 the billing cycle is 35 days and the interest rate on the credit card is 16 How much do you owe O 304 60 O 306 40 0000 CO

Suppose a parabola has equation 24 y 3 x 8 1 The vertex is at The focus is at and directrix is

10 9 8 7 6 5 4 The focus F is at 1 2 486 7 65 721 2 3 5 7 8 9 10 The directrix has equation y 4 7 8 9 10 The vertex V is at 1 1 The line of symmetry has equation 1 symmetry The equation of the parabola is x 1 2 12 y 1 0 Enter the equation of the directrix o Enter the equation of the line of Enter the equation of the parabola The z intercept s are 2 46 0 4 46 0 there is more than one separate them with commas if none exist enter DNE The y intercept s are 0 0 92 Enter the exact coordinates of the x intercept s if Enter the exact coordinator

8 8 Say that a helicopter is hovering 1 2 mile above a road It tracks a car moving away from it and measures that the distance z between it and the car is changing at a rate of z t 65 mph at the moment that the distance is z 1 mile IN a State a formula that relates the variables x and z b Find the horizontal speed dx dt of the car at the moment that z 1

5 Throughout this problem you need to use Calculus for your solu tions No credit will be given for reading the results off of a graph All solutions must be exact that is not decimal approximations x x Let h x x x 2 x 1 5 2x which in factored form is h x a Find all critical points of h x Recall that the quadratic for mula is x b b 4ac 2a b Determine the interval s on which h x is increasing and de creasing c Determine whether each critical point is a relative maximum a relative minimum or neither

Consider the ellipse with equation The semimajor axis has length 5 The semiminor axis has length 3 The focal length is 4 5 3 The vertices are located at 4 5 4 15 separated by commas The foci are located at 4 6 4 14 y 10 5 X G 1 enter the coordinates of each vertex enter the coordinates of each focus separated

Suppose a parabola passes through the point 55 horizontal directrix The directrix will have equation The equation of the parabola will be 61 has focus at 7 6 opens upward and has a Enter the equation of the directrix Enter the equation of the parabola

Use the power series for f x arctan x to approximate the value using R 0 001 Round your answer to three decimal places N arctan

x 11 Let f x 24x 3 a Find the critical value s of f x b Find the interval s on which f x is increasing decreasing c Determine whether the critical value s are local maxima local minima or neither d Find the interval s on which f x is concave up concave down

For all Taylor polynomials Pn x that approximate a function f x about x a Pn a f a False True 11

Find the derivatives of the following functions You do not to simplify your answers a f x ln x x 3t 1 b g t 4 sin t c h y tan y 2 y 6 r tan 2x

2 Let f x x 2 Show that f x 2x using the definition of the derivative No credit will be given for using the short cut rule

If y then all line segments comprising the slope field will hae a non negative slope O True

5 25 3 For 5 5 24 10x x Part A Determine if long division will be used for this integration and justify your answer 5 points Part B Simplify the integrand and use a u substitution to rewrite the integral with new limits of integration 15 points Part C Evaluate the integral Round to the nearest thousandth 10 points dx

Use the definition of the Taylor series to find the first four non zero terms of the Taylor series about 7 5 for f x

the power series Cn x 1 diverges for x 2 then it diverges for x 5 O True False

A cone will be inscribed in a sphere of radius 10 in Part A Find the exact dimensions of the cone of largest volume that will fit inside the sphere 20 poin Part B Find the maximum volume cone to the nearest tenth 20 points

Suppose s t t t on the interval 0 3 What is the displacement on 0 3 O 6 O 3 O 3 O 6

Convert x y 5 25 to polar form X Drag and drop an answer to each box to correctly complete the equation r 10 5 5 10 sin cos

7 1 Determine the volume and surface area of the closed cylinder below Surface Area 2rr 2nrh Volume Base area h MARKED BY D 7 2 10cm INFORM A rectangular box has a volume of 2400 cm Its length 3 times the height The width is 2 cm less than the height Let the height be x Calculate 7 2 1 the height of the box 14cm of the plas 3 Val 4

8 Solve for x 2x 10 48 9 Solve for x 102 38

Find a power series for the function centered at c 3 c 2 2x 1 f x n 0 Determine the interval of convergence Enter your answer using interval notation

3 Analyse and then sketch the function 3x x 7x 2 a Determine the asymptoles b Determine the end behaviour and the Intercept Find the critical points and the points of inflection d Determine the interval of the increasing I decreasing and th Con Cavity es sketch the graph

APPLICATIONS 39 Leaning Ladder A 20 ft ladder is leaning against a building If the base of the ladder is 6 ft from the base of the building what is the angle of elevation of the ladder How high does the ladder reach on the building 40 Angle of the Sun A 96 ft tree casts a shadow that is 120 ft long What is the angle of elevation of the sun 41 Height of the Space Shuttle An observer views the space shuttle from a distance of 2 mi from the launch pad a Express the height of the space shuttle as a function of the angle of elevation 8

Find a power series for the function centered at c and determine the interval of convergence f x 3x 2 c 6 Step 1 Rewrite the original function f x f x 3 3x 2 18 18 20 3x 20 3 3 3 after adding and subtracting 18 in the denominator as follows 3x 2 x 6

7 Path of a Projectile If a projectile is fired with an initial speed of u ft s at an angle a above the horizontal then its position after t seconds is given by the parametric equations y sin a r 161 x v cos a t where x and y are measured in feet Show that the path of the projectile is a parabola by eliminating the parameter 1 8 Path of a Projectile Referring to Exercise 37 suppose a gun fires a bullet into the air with an initial speed of 2048 ft s at an angle of 30 to the horizontal a After how many seconds will the bullet hit the ground b How far from the gun will the bullet hit the ground e What is the maximum height attained by the bullet

k x x 5 x 3 16 Step 3 of 3 Find two points on the graph of the parabola other than the vertex and x intercepts Answer Any lines or curves will be drawn once all required points are plotted Step 1 Vertex 1 0 A B 10 S S 40 Enable Zoom Pan 5 KA 10

Find the surface area of each figure 12 ft 3247 0 Ob 2145 Oc 2589 1232 13 ft 13 ft 10 ft 19 ft 9 ft YLL

Describe the possible lengths of Pine Avenue Pine Avenue must be longer than miles and shorter than miles

2 Find the global maximum and minimum of over the domain f x y 2x y2 4x D x y R x y 4 y x

sider the given function and the given interval f x x 3 1 4 a Find the average value fave of f on the given interval fave b Find c such that fave f c smaller value larger value C C c Sketch the graph of f and a rectangle whose area is the same as the area under the graph f x 4 f x 4

Find the area of the region bounded by the parabola y 5x the tangent line to this parabola at 4 80 and the x axis

Question 1 of 16 Step Identify the function for the given graph Answer of x x Of x Correct 21 12 x Of x x f x x Of x 1 4 Of x x of x x

Evaluate the determinant of the matrix A 3ac 5ad 3bc 3bd

Use Cramer s rule to solve the system Answer 9w 27x 18y 108 w 6x y 24 x z 72 y z 0 If there is a solution write your answer in the format w x y z Selecting an option will display any text boxes needed to complete your answer No Solution

Evaluate the determinant of the matrix A 25 0 37 49

A musician is playing a tuba If she plays a note and holds it for a significant length of time the air pressure near the opening of the tuba can be modeled by th following function p 1 101 325 4 0 1 cos 290 6 In this equation p t represents the air pressure in pascals and is the time in seconds Find the following If necessary round to the nearest hundredth Frequency of p cycles per second Time for one full cycle of p seconds Maximum air pressure pascals

Use the graph on the right to answer the following questions a Estimate the pressure at altitudes of 9 000 feet 18 000 feet and 29 000 feet b Estimate the altitudes at which the pressure is 23 18 and 13 inches of mercury c Estimating beyond the boundaries of the graph at what altitude do you think the atmospheric pressure reaches 5 inches of mercury Is there an altitude at which the pressure is exactly zero Explain your reasoning a The pressure at 9 000 feet is inches of mercury the pressure at 18 000 feet is of mercury Round to the nearest integer as needed Atmo pressure in of mercury 35 30 25 20 15 10 5 0 32 0 5000 24 10000 21 20000 15 30000 11 10 000 20 000 30 000 Altitude feet inches of mercury and the pressure at 29 000 feet is in

Given x is number of items Demand function d x Supply function s x 4 x Find the equilibrium quantity 1225 4900 OS Find the producer surplus at the equilibrium quantity 1286785 it

Find the equation of the parabola with the following properties Express your answer in standard form Answer 2 Points Focus at 3 4 Directrix is the x axis

e the product to sum identities to rewrite the following expression as a sum or difference 4sin 55 cos 55 swer 2 Points

Use the information in the diagram to determine the radian measure of the angle 0 5 0 5 0 5 Enable Zoom Pan p 05 XA

Find the perimeter of the triangle whose vertices are 2 5 3 0 and 1 Answer 2 Points 4 Write the exact answer Do not round

Solve the following system by the inverse matrix method if possible If the inverse matrix method doesn t apply use any other method to determine if the system is inconsistent or dependent If there is a solution write your answer in the format x y z Answer 3x 2y 4z 15 x 4y 3z 4 4x 2y 7z 11 Selecting an option will display any text boxes needed to complete your answer Keyp Keyboard Short

0 3 7 9 1 2 4 5 6 1 1 5 lim f x 1 lim f x 1 x 5 7 6 5 f 5 DNE lim f x DNE x 5 2 lim f x 2 8 f 2 DNE lim f x 2 x 2 lim f x DNE lim f x 0 x 0 lim f x 0 4 0 3 2 1 11 12 13 14 15 17 18 19 6 20 5 4 3 2 1 2 3 4 5 6 0 1 2 13 lim f x ONE 2 0 f 0 ONE lim f x x 1 16 f 1 ONE 5 4 lim f x x 1 lim f x DNE x 1 lim f x x 3 lim f x 2 x 3 lim f x x 3 f 3 a x 21 22 23 24 25 26 27 lim f x 1 x 4 lim x 4 f x 1 lim f x ONF x 4 4 ONE lim f x 1 H 5 lim f x 1 x 5 lim f x ONE x 5 28 f 5 NE 29 List the value s of r at which f is discon tinuous x 1 3 x 5