Application of derivatives Questions and Answers

Emi computes the mean and variance for the population data set 87 46 90 78 and 89 She finds the mean is 78 He steps for finding the variance are shown below 2 87 78 2 46 78 90 78 78 78 89 78 9 2 32 2 12 2 0 11 5 81 1 024 144 0 121 5 5 678 5 135 6 What is the first error she made in computing the variance Emi failed to find the difference of 89 78 correctly Emi by N instead of N

Complete the following truth tables using the given values of p q and r Pq r FT F Pr Pq r FTF O a T pv q ni a 9 r pr V gr pVg r

Note sin x cos x 1 Take the derivative of each trigonometric function a y 4sin x 5cos x 4 b y sin 3x cos 3x

1 0 1 15 0 1 1 14 1 1 0 16 17 0 0 68 17 9 11 2 0 9 0 65 nits a consistent and cancel throughout the calculations

24 g t t 4 h t t 2 Find 5g t 3h t

Get a simila A baseball team p average attendan 32000 Assume th

Compute the magnitude of the tangent vector and the unit tangent vector at t 1 for each of the following vector functions Provide a sketch of the curve from t 0 to t 1 a r t t sin t cos t b s t t t cos 2 t t sin 2t c u t t sin 100xt t t cos 100nt

Suppose you make a deposit of 200 at the beginning of every year value of the account in dollars can be found by the given equation 200 200r 200 200 1 r 200 300r

37 44 a Find the vertical and horizontal asymptotes b Find the intervals of increase or decrease c Find the local maximum and minimum values d Find the intervals of concavity and the inflection points e Use the information from parts a d to sketch the graph of f 37 f x x 2 x 1 38 f x Ir x 2 2

3 Suppose that 3 f x 5 for all values of x Show that 18 f 8 2 30

4 5 Use the given graph of f to find the following a The open intervals on which f is increasing b The open intervals on which f is decreasing c The open intervals on which f is concave upward d The open intervals on which f is concave downward e The coordinates of the points of inflection 5 y 4 ya 1 0 1 X 1 0 1 X

41 54 Find the absolute maximum and absolute minimum values of f on the given interval 41 f x 12 4x x 0 5

dy Use implicit differentiation to find and then dx y 9e 10x dy dx 11 d y

The radius of a circle is increased from 2 00 to 2 04 m a Estimate the resulting change in area b Express the estimate as a percentage of the circle s original area a The estimated change in area is Type an exact answer using x as needed b The estimated change in area is of the circle s original area Type an integer or a decimal Do not round

A weight is attached to a spring and reaches its equilibrium position x 0 It is then set in motion resulting in a displacement of x 14 cos t where x is measured in centimeters and t is measured in seconds See the figure shown to the right Answer parts a and b a What is the spring s displacement when t 0 0cm Type an integer or decimal rounded to one decimal place as needed What is the spring s displacement when t cm Type an integer or decimal rounded to one decimal place as needed What is the spring s displacement when t 3x 4 cm Type an integer or decimal rounded to one decimal place as needed b What is the spring s velocity when t 0 cm sec Type an integer or decimal rounded to one decimal place as needed What is the spring s velocity when t cm sec Type an integer or decimal rounded to one decimal place as needed 3x What is the spring s valocitu who NECED

3 Use transformations to graph the function b x a The parent function of b x b The factor of in the function indicates a is y vertical or horizontal shrink or stretch c For any point x y on the graph of the parent function the point x y is on the graph of b x x 1 x d Choose 5 convenient points on the given graph of the parent function and transform the points appropriately to graph b x on the same coordinate system Use the provided table to guide your work 1 points on the graph of the parent function in the form 2 12 8 8 cu 0 transformed points on the graph of y b x in the form

The equation below gives the position s f t of a body moving on a coordinate line s in meters t in seconds Find the body s velocity speed acceleration and jerk at time t 4 S 7 9 sin t X The velocity of the body at t X The speed of the body at t sec ism sec X The jerk of the body at t 4 sec ism sec X The acceleration of the body at t 4 sec ism sec sec ism sec ALLD sec

dy Write the following function in the form y f u and u g x Then find dx y Write the given function in the form y f u and u g x Choose the correct answer below O A y f u u and u g x 6 OB y f u u and u g x 3x X OC y f u u and u g x OD y f u u and u g x Find dx dy dx as a function of x 6 3x 3x 3x 3 X 3x as a function of x 3 X

The equation for free fall at the surface of some planet s in meters t in seconds is s It takes sec for a rock falling from rest to reach a velocity of 25 1 m s Round to the nearest tenth as needed s 8 9712 How long does it take a rock falling from rest to reach a velocity of 25 1 m s on this planet CIT

K Use implicit differentiation to find 5x y 4xy 6 dy dy dx

The graph of the accompanying figure shows the average annual percentage change y f t in a particular country s gross national product GNP for the years 1978 1983 Ay 5 10 8 P 6 4 2 0 1978 1979 1980 1981 1982 1983 Year Q G Choose the correct graph of O A Ay 10 J 1978 10 O C Ay 10 1978 10 1983 1983 Q 5 dy dt OB 10 10 1976 O D 10 Ay 10 Ay 1976 1983 1983 Q a

A dinghy is pulled toward a dock by a rope from the bow through a ring on the dock 7 ft above the bow The rope is hauled in at the rate of 3 fu sec Complete parts a and b a How fast is the boat approaching the dock when 25 ft of rope are out sec Type an integer or a simplified fraction b Al what rate is the angle 0 changing at this instant radisec Type an integer or a simplified traction

y 10 x 1 10 y 0 Choose the correct graph of the curve and the tangent line below A AY 20 1 1 10 5 0 0 10 Question Viewer OB Ay 20 2 10 0 the tangent line together 0 10 Q Q KIIB O C Ay 20 10 0 0 10 Q G

Find the point y at which the graph of y 5x 2x 2 has a horizontal tangent m The function y 5 2x 2 has a horizontal tangent line at Type an ordered pair Simplity your answer

Score 3 4 3 4 answered Question 3 A rectangular field whose area is 19000 square feet is to be divid used to enclose the field as well as for the internal partitions length width

A quadratic function is given f x 5x 20x 1 a Express fin standard form b Sketch a graph of f 20 10 30 20 10 10 20 y 30 30 10 20 X 20 10 30 30 10 10 30 20 30 y

Find the maximum or minimum value of the function f x 36x x Is this a maximum or minimum value O maximum value O minimum value

29 s t 3t4 4t 6t y 1 y y 1 31 g y

Find the reference angle for the angle 47 6 The reference angle is Type your answer in radians Type an integer or a simplified fraction

Find the exact value of the following expression Write the answer as a single fraction Do not use a calculator sin 23x 4 COS 5 6 COS 23T 4 sin 5 6 23T 5x 23x 5T sin COS cos sin 4 6 4 6 Simplify your answer Type an exact answer using radiolo

Find the exact value of the trigonometric function of 0 from the given information 4 5 cos 0 8 in quadrant IV find tan 0 tan 8 Simplify your answer including any radicals Use integer or

mplete the following statement Let P x y be a point on the terminal side of an angle 0 in standard position Let r sin 8 y Then sin 8 cos and tan 0

the exact value of the expression Do not use a calculator sin sin H R 6 COS cos 0 Simplify your answer Type an exact answer using radicals as needed Rationali www

Use a calculator to find the approximate value of the expression Round the answer to two decimal places 5x 12 sin 5x 12 Round to two decimal places as pooded sin

Determine the missing coordinate of the point that lies on the graph of the unit circle in quadrant III The missing coordinate is y Simplify your answer including any radicals Use integers or fractions for any numbers in the expression

17 24 Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f Use the graphs and of Sections 1 2 and 1 3 transformations 17 f x 3x 1 x 3

Evaluate the integral 29 x x 16 S dx A Which trig substitution is correct for this integral Or 16 sin 0 Ox 4 tan 0 Ox 16 tan 0 0x 29 sec 0 Or 16 sec 0 Or 4 sin 0 B Which integral do you obtain after substituting for and simplifying Note to enter 0 type the word theta do C What is the value of the above integral in terms of C D What is the value of the original integral in terms of

A table of values for a linear function is given Fill in the missing value X y 1 0 m 3 8 Calculate m 1

Using the piecewise function defined below determine the indicated function value 5 2x if x 3 2x x if 3 x 5 x 71 if 5 x g x a g 4 b g 0 5 c g 5

Expand the expression x x 1 4x 8

OT inequ 10 9 00 T 9 y x 3 5 y x 2 4 t f 10 9 8 7 99 H doi 5 6 B 8 9 10 x

Use FIRST PRINCIPLES to determine a general expression for the slope of a tangent line to g x x 5x 10

2 Express the repeating decimal as a fraction 0 44444

5 Find the linear approximation of the function f x 1 x at a 0 and use it to approximate the numbers 0 9 and 0 99 Illustrate by graphing f and the tangent line

A particle is moving on a vertical line in meters so that its coordinate at time is y t 12t 3 a What is the velocity after 3 seconds FG 1211 0 3 12 V87 31 12 3 3 3 12 15 b When is the particle at rest 31 12 3 37 9 4 400 311330 37 3 4 c When is the particle moving in a positive direction d Find the total distance traveled from 0 t 3 e Find the acceleration at timer and after 3 seconds a t v 01 9 1 317T 96 6 3 18 f When is the particle speeding up When is it slowing down

1 D anapo Use the equation y x 4 to answer parts a through g belov g Find the domain and range of the function What is the domain Select the correct choice below and if n the answer box to complete your choice OA The domain is Type an inequality or a compound inequality B The domain is all real numbers below and if n

I in the blank with the appropriate word hen solving a system of equations if a true state

Chapter 2 Test Page 8 of 10 b The graph of the equation y 9ry is shown below 5 v 4 3 2 1 5 4 3 2 11 2 3 4 5 I 1 2 3 4 5 1 March On the graph above draw the tangent line at the point 4 2 c Find the equation of the line that is tangent to the graph at the point 4 2

c The derivative of a polynomial is a polynomial TRUE FALSE

As on the previous page f x 4x 7x 2 c Find the equation of the line tangent to f x at x 1 d Find the equation of the line normal to f x at x 1 e At which values of a is the tangent line horizontal