Differentiation Questions and Answers

ST Find 10 2 if sin x

f t 9t 1 5 If find f t Simplify fully no negative exponents no fractional exponents n exponents under radicals 6 points chan fok

Compute lim Ak where A k selectes 1 2 1 2 2 3 1 3

3 For f x 2x 16x 5 find the following A f x B The slope of the graph of f at x 2 C The equation of the tangent line at x 2 D The value s of x where the tangent line is horizontal PODER PUPAS A f x B At x 2 the slope of the graph of f is C At x 2 the equation of the tangent line is y D The tangent line is horizontal at x Use a comma to separate answers as needed somligar www

y cscx B phase shift if any and c range of the following function 1 a The period of y CSC X is 2 Simplify your answer Type an exact answer using as needed Use integers or fractions for any numbers in the expression

Find d dx 2 d 2x dx 3 2x 3 5 2 7x 5 2 7x 7

Use the graph of the function f shown to the right to estimate the indicated function values and limits Complete parts A through E lim f x M A Find the value of lim f x Select the correct choice below and if necessary fill in the answer box to complete x 3 your choice OA x 3 Z OB The limit does not exist B Find the value of lim f x Select the correct choice below and if necessary fill in the answer box to complete

Use the four step process to find f x and then find f 1 f 2 and f 3 f x 2x x 1 f x f 1 f 2 f 3 Type an integer or a simplified fraction Type an integer or a simplified fraction Type an integer or a simplified fraction

Find the critical number of the function f x 3x 7x X

Does there exist a continuous function f x such that f 0 6 f 2 10 and f x 7 for all a in 0 2

Find the absolute maximum and absolute minimum values of the function on the interval 2 5 4 Enter 1000 for any absolute extrema that does not exist Absolute maximum Absolute minimum 6 f x 26 X

Consider the function f x on the interval 3 6 A Find the average or mean slope of the function on this interval Average Slope B By the Mean Value Theorem we know there exists a c in the open interval 3 6 such that f c is equal to this mean slope Find all values of c that w and list them separated by commas in the box below List of values

Find the most general antiderivative of the function Check your answer by differentiation Use C for the constant of the antiderivative F x f x x 5 5x 15

Since raindrops grow as they fall their surface area increases and therefore the resistance to their falling increases A raindrop has an initial downward velocity of 10 m s and its downward acceleration is a 19 0 9t if 0 st 10 10 if t 10 If the raindrop is initially 500 m above the ground how long in seconds does it take to fall Round your answer to two decimal places S

Consider the following problem a box with an open top is to be constructed from a square piece of cardboard 3 feet wide by cutting out a square from each of the four corners and bending up the sides Find the largest volume that such a box can have a Draw several diagrams to illustrate the situation some short boxes with large bases and some tall boxes with small base Find the volume of each configuration Does it appear that there is a maximum volume If so estimate it b Draw a diagram illustrating the general situation Let x denote the length of the side of the square being cut out Let y denote the length of the base c Write an expression for the volume V in terms of both x and y V d Use the given information to write an equation that relates the variables x and y e Use part d to write the volume as a function of only x V x f Finish solving the problem by finding the largest volume in ft that such a box can have V

A steel pipe is being carried down a hallway that is 9 ft wide At the end of the hall there is a right angled turn into a narrowe hallway 6 ft wide What is the length of the longest pipe in ft that can be carried horizontally around the corner Round yo answer one decimal place ft 0 6

Evaluate Be sure to check by differentiating 1 6 3x dx 6 3x dx x 2 xi 16 3x dx 0 Type an exact answer Ilse paronth

Calculate all four second order partial derivatives of the function 3x f x y y 0 and confirm that the mixed partials are equal Y fax x y fay x y

Calculate all four second order partial derivatives of the function f x y 9xe and confirm that fry fyr Assume the variables are restricted to a domain on which the function is defined fxx fay fyx fam

Find the length and direction when defined of u xv and vxu u 8i 3j 5k v 4i 4j 2k uxv 6 17 Type an exact answer using radicals as needed Select the correct choice and fill in any answer boxes in your choice below A 7 3 17 2 3 17 Type exact answers using radicals as needed The direction of u x v is OB The direction of u xv is undefined i j vxu Type an exact answer using radicals as needed 10 3 17 k

What is the energy in joules of one photon of microwave radiation with a wavelength 0 122 m O 1 63 x 10 24 J 4 07 x 10 10 J O 2 70 x 10 43 J 5 43 x 10 33 J

Find the length and direction when defined of u xv and vxu u 8i 3j 5k v 4i 4j 2k luxv Type an exact answer using radicals as needed

Graph the logarithmic function f x log x 4 Drag the two blue points on graph to the correct points that f x passes through Then use the remaining red point to drag the vertical asymptote to the correct line Note that the points drag in increments of fourths

Use the properties of logarithms to expand the following logarithm as much as possible Rewrite as a sum difference or product of logs log r y r y7

The shape of a colony of bacteria on a Petri dish is circular Find the approximate increase in its area if its radius increases from 10 mm to 19 mm Recall that the area of a circle is A

A cube with sides 9 inches long is covered with a coat of fiberglass 0 6 inch thick Use differentials to estimate volume of the fiberglass shell THE The volume of the fiberglass shell is approximately in

The total cost in dollars of manufacturing x auto body frames is C x 40 000 200x A Find the average cost per unit if 50 frames are produced B Find the marginal average cost at a production level of 50 units C Use the results from parts A and B to estimate the average cost per frame if 51 frames are produce

dy Differentiate implicitly to find Then find the slope of the curve at the given point dx 6x 7y 157 5 1 dy dx The slope of the curve at 5 1 is 11

Suppose that the price p in dollars and the number of sales x of a certain item are related by 4p 4x 2px 72 If p and x are both functions of time measured in days Find the rate at which x is Changing when x 3 p 6 and dp dt 1 1 The rate at which x is changing is Round to the nearest hundredth as ne day s per sale day s sale s sale s per day

uppose that x and y are related by the given equation and use implicit differentiation to determine x y y x 6 dy dx dy dx

Find all points where the partial derivatives of f x y are both 0 where f x y x 12x y 27y Number of point s where the partial derivatives of f x y are both 0 Choose one

Find the exact solutions of the equation in the interval 0 2 sin 2x cos x 0 X X X X X X X smallest value largest value

1 Given a function of f x 2x 8x 3 on 1 5 answer the following questions You cannot use the graphic calculator to find the answers ing que A Find the critical point s within the given interval mo 8 B Determine the absolute max and absolute min of f x within the given intenol

ven a function f x 6x 3x find the antiderivative F x that satisfies F 1 3 7 10t 5 and g 1 10 find g t

19 36 Sketch the region enclosed by the given curves and find its area 19 y 12x y x 6 20 y x y 4x x 21 x 2y x 4 y 22 y x 1 x y 1 23 y 2x y x 24 y x y x 25 y x y x 0 x 16 26 y cos x y 2 cos x 0 x 2 27 y cos x y sin 2 r

a 4 pts What is the average rate of change of f between 3 and 2 Write the answer as a fraction no decimals or mixed numbers b 4 pts What is the equation of the secant line through 3 f 3 and 2 f 2 Use fractions in the equation no decimals or mixed numbers Sketch a graph of the function and this secant line 5 8 pts Sketch a graph of y 5e 7 What is its y intercept and what is the equation of its horizontal asymptote Fill in the blanks for the end behavior below too

A cosmetic company gives sales training to its newly hired employees To determine how effective the training is the firm compared the monthly sales of a group that has completed the training with a group that has not The numbers to the right indicate thousands of dollars of sales last month Represent the two sets of data on a single stem and leaf display Does the orientation program seem to be succeeding Choose the correct answer below OA With Training No Training 51348 96212022234689 O C 99862337 8333247 With Training No Training 3481 5 02223468929621 37399862 7483332 With Training No Training 29 32 36 43 48 43 22 38 42 39 39 21 26 15 43 26 28 14 13 20 22 29 22 24 33 47 23 22 18 37 XX B With Training No Training 51348 9621 2022234689 99862337 7483332 D With Training No Training 833321348 96212022234689 99862337 74 5 Does the orientation program seem to be succeeding OA The orientation program is not working because employees who received training had lower sales values OB The orientation program is working because employees who received training had higher sales values OC The orientation program is working because employees who received training had lower sales values OD The orientation program is not working because employees who received training had higher sales values

A car comes to a stop six seconds after the driver applies the brakes While the brakes are on the following velocities are recorded Time since breaks were applied seconds 0 1 5 3 4 5 Velocity feet per second 63 34 12 0 Give lower and upper estimates using all of the available data for the distance in feet the car traveled after the brakes were applied Lower Upper

Find the following antiderivative 7x 3x 2 dx Do not include the constant C in your answer For example if you found the antiderivative was 2x C you would enter 2 x

5

5 9 X 0

second derivative of the functions Wiritize in the variannes a y 2 74 44 5 1 S xy y x 0

es e O Rama produces two bike models Voltage and EasyRider Departmental overhead data follow Department Fabricating Assembly Budgeted Cost 97 200 80 000 Allocation Base Machine hours MH Direct labor hours DLH Required 1 Compute departmental overhead rates using a machine hours to allocate budgeted Fabricating costs and b direct labor hours to allocate budgeted Assembly costs 2 The company reports the following actual production usage data Compute the overhead cost per unit for each model Machine hours per unit Voltage 2 0 MH 1 6 DLH Direct labor hours per unit 3 The company reports additional information below For each model compute the product cost per unit Direct Materials Budgeted Usage 8 100 MH 1 600 DLH EasyRider 3 0 MH 0 5 DLH 91 81 Per Unit Selling Price Voltage EasyRider 261 151 4 For each model compute gross profit per unit selling price per unit minus product cost per unit Direct Labor 46 16 Complete this question by entering your answers in the tabs below

10 Given y f x CSC cot 1n 9x find dy dx

ind the absolute maximum and minimum of f x y 3x 3y within the domain x y 4 1 Absolute minimum of f x y is 2 Absolute maximum of f x y is 4 64 4 64 X 8 4852813742386 x 8 4852813742386

tan 15x 4 1 0 4 810 5N Alm 4x 3 O 0 1 0 1 W SPE 4 19 1 0 4 Type an exact answer usin

K I Determine what fraction of the circumference of the unit circle the value of s represents For exa 1 represents of the circumference of the unit circle Do not use a calculator 2 I represents of the circumference of the unit circle 4 Type an integer or a simplified fraction

1 represents of the circumference of the unit circle

39x 1 x 9 Simplify your answers Type an integer or a fraction C 39x

angle of 360 has an equivalent radian measure of C