Application of derivatives Questions and Answers

1 4 Composition of Functions Express the function h x x 5 in the form h x fog x where f x x and g x 1 a g x b f x Question Help D Post to forum Submit All Parts

ample 18 On what interval s is the function g x Show solution h t is concave down on h t is concave up on Exercise 1 5 9 On what interval s is the function h t t 3 2 concave up and down Hint Recall the graph of f x x Show graph Important Topics of this Section 2 x 1 Transformations 3 increasing and decreasing

Consider the function g x 1 as a reference There are 3 steps T function f x a Select an answer 1 b Reflect Select an answer Select an answer c Shift Select an answer by 5 Describe its transformation in words using the tool kit by

Set up but do not evaluate integral expressions for a the mass b the center of mass and c the moment of inertia about the z axis the solid enclosed by the cylinder y x and the planes z 0 and y z 1 p x y z 3 x 1 a the mass m b the center of mass x y z HE Iz 1 m 1 m 3 1 x2 x c the moment of inertia about the z axis dz dy dx dz dy dx dz dy dx dz dy dx dz dy dx

Consider the function f x 2x x 4x 3 Find the average rate of change of this function on the interval 4 2 By the Mean Value Theorem we know there exists a c in the open interval 4 2 such that f c is equal to this average of change Find the two values of c in the interval which work enter the smaller root first

f x 9x x 3 A Find all critical numbers of f If there are no critical numbers enter NONE Critical numbers B Use interval notation to indicate where f a is increasing Note Use INF for co INF for o and use U for the union symbol Increasing C Use interval notation to indicate where f x is decreasing Decreasing D List the x coordinates of all local maxima of f If there are no local maxima enter NONE values of local maxima E List the x coordinates of all local minima of f If there are no local minima enter NONE values of local minima F Use interval notation to indicate where f x is concave up Concave up G Use interval notation to indicate where f x is concave down Concave down H List the values of all inflection points of f If there are no inflection points enter NONE

Clotis ution is one to one Find the inverse of the

Consider the function f x on the interval 2 11 Find the average rate of change of the function on this interval By the Mean Value Theorem we know there exists a c in the open interval 2 11 such that f c is equal to this average rate of change For this problem there is only one c that works Find this value of c

Suppose f x is continuous on 4 8 and 2 f x 2 for all x in 4 8 Use the Mean Value Theorem to estimate f S f 4 Answer f 8 f 4

Consider the function f x 2x 5 List the values of the inflection points of f If there are no inflection points enter N a

Consider the function f x x 4x 2 on the interval 0 4 Verify that this function satisfies the three hypotheses of Rolle s Theorem on the inverval f x is f x is and f 0 f 4 on 0 4 on 0 4 Then by Rolle s theorem there exists at least one value c such that f c 0 Find all such values c and enter them as a comma separated list Values of c

Graph the function as a blue line for cuno and then d

The rate of growth of a microbe population is given by m x 30xe 2x where x is time in days What is the net growth between day 3 and day 7 111 306 789 117 238 789 222 613 544 222 613 533 SHEE

Evaluate using integr Sx xe8x dx xe8x e8x 8 Xe8x 8 8x 64 C C xe8x C e8x 8x 1 C

The critical numbers of the function and t Twitte f t 912 3 15 3

Find all critical values for the function and then list them separated by commas in the box below List of critical numbers 3r 7 1

Consider the function f x 3 6x The absolute maximum value is and this occurs at a equal to The absolute minimum value is and this occurs at a equal to 3 x

The function f x 3x 7 e has one critical number Find the critical numbe 3 17 6 2e

Evaluate each of the following indefinite integrals a f 3x 5 dx

The surface of an island is defined by the following function over the region on which the function is nonnegative Find the volume V of the island shown to the right 2 x x 12 8 2 I lo In

Find the value s of c such that the area of the region bounded by the parabolae y x c and y c x is 576 Answer separate by commas c

Find the volume of the solid that results when the red region is revolved about the x axis f x 5 x a 2 b 5 click on image to enlarge C yaf u ymf u b

Find the area of the region between the curves y x 1 y 9 x x 1 and x 2 Area between curves

1287 Selective Laser Sintering The SLS machine needed the wall vibration damped out below 1 in 1 s Which damping function would best fit this situation A v t 2e 02tsint B v t etsin2t C v t 2e 04tsin2t D v t 4e 08tsin2t Select one A B O C D Incorrect

Help Lecture 57 Ex 57 3 15 Find the three third roots of 64cis24 57 Select one 4cis248 4cis8 4cis98 4cis128

Help Lecture 79 Ex 79 2 13 Find the distance between Dogfish Reef and Prince Rupert British Columbia Round to 1 d p 53 2 mi 59 5 D 58 3 mi ce Ruper

Igor purchased a car for 30 000 He made a down payment of 17 500 and borrowed the remainder at an interest rate of 3 5 compounded continuously for 4 years Estimate his monthly loan payments for the 4 year period rounded to the nearest dollar elect one 5 320 300 280 x Incorrect

If m f x M for a x b where m is the absolute minimum and M is the absolute maximum of f on the interval a b ther b S f Ja m b a Use this property to estimate the value of the integral 3 3 1 17 x2 9 1 x 0 f x dx M b a dx smaller value larger value 4

sin r 1 3 T 4 4 dx is equal to 2 x log sin x 7 c 4 x x 7 x log cos x c 4 2 x log sin x 7 c 4 x log sin x 30 4 c

5pc Find g x if g x 96 Select one O O O O cosx sinx X cos xsing x xcosx sinx x xcosx sinx x4 X

1285 Decompose the ratio Select one 7 1 x 5 1 x 5 1 x 4 1 x 4 1 x 7 1 x 4 1 x 3 1 x 3x 11 1 x into a sum of partial fractio

894 Find y if y 4x5 2x x 2x 7 Select one O O O O y 80x 48x 6x y 240x 48x6 y 80x 24x 6x y 80x 12x 6x 2 20x424x 6x y

Graph the parametric equations x 4 t y 21 in the specified parameter interval Use the standard viewing 25152 Choose the correct graph below All graphs are displayed in viewing windows 10 10 by 10 10 Select one O a O b OC Od

140 Is f x Arcsec x a function or a relation If it is a function use interval notation to writ its domain and range Select one function domain 1 and 1 function domain 1 and 1 range 0 2 and 2 relation function domain range 0 2 and 2 1 and 1 range 0 2 and 2

units Help Lecture 98 Ex 98 2 498 The amount of force F applied versus time followed the pattern F t 3t 12t 24 Find the time at which the force was at a maximum In other w find when F t 0 Units aren t required

X x 1 X A rancher has 984 feet of fencing with which to construct adjacent equally sized rectangular pens as shown in the figure above What dimensions should these pens have to maximize the enclosed area y Maximum area S ft help numbers ft y ft2

elp Lecture 92 Lesson 92 and Ex 92 1c 992 Use definite integrals to find the exact area under the curve over the interval 1 7 f x v3

The scalar kis equal to 4 Vectors v and V are defined as V 3 5 and V 2 1 Evaluate kv1 V Select one O a 1 4 O b 10 19 O c 5 1 O d 4 16

Express the quotient 2 2 in trigonometric form when z 1 i and 3 i Select one O a O b O c COS isin cos isin co 19m 12 i sin 1927 Od con sin 127 d 19 19T COS 12

The graph of the polar equation r a where a is a constant is Select one O a a limacon O b a circle O c a horizontal line O d a vertical line

Help Lecture 87 Ex 87 2 787 For b t 10e Ostsin 8t what is the minimum value of t for which b t damps out to va less than or equal to 2 Round to 2 d p

1472 Which of the following is the symbolic form of the cosecant function shown in the graph x axis tick marks are 1 2 radians apart Select one A A f x CSc X T1 2 B f x CSc x 4 C f x csc x 2 D f x csc x 1 TU

elp Lecture 86 Ex 86 1 Medical Vital Signs The patient s daily body temperature fluctuated according to following pattern 10 86 T h 98 2 sin n F If h is in hours find patient s body temperature at h 18 hours No GC

13 9 Find the missing parts Round to 1d p Side Z Not a solution for Side Z Angle x Angle y Not a correct angle Check Response hist Choose Choose 12 6 12 4 43 8 12 Q X 12 Z 72 y 9

9 86 Volcanism The geyser erupted according to a predictable daily pattern of pressure buildup of the form Select one 2 4 hr 4 2 hr 6 8 hr P t 110 30 sin F 8 0 t 24 hours If the geyser erupted at pressures above 130 atm what was the duration of a typical eruption

395 Find the derivative of f x x In x Select one O O O x 1 3lnx 3x Inx x 3 lnx x 1 3lnx

Here are the meanings of some of the symbols that means is a subset of C means is a proper subset of means is not a subset of is the empty set For each statement decide if it is true or false Statement 1 2 3 4 5 C 1 2 4 11 13 15 11 12 13 14 15 16 21 23 26 27 28 21 23 26 27 28 C g k m Tr O

22 5 mi 18 7 mi Help Lecture 77 Ex 77 2 12 7 Volcanism Find the distance between Mount St Helens S and Mount Adams A in Washington if the distance between Gilbert Peak G top right vertex and Mount Ada A is 19 8 miles Round to 1 d p 51 2

6 Find the derivative of f x 5x x sin x 3 Select one f x 20x 3x 4 cosx 3e f x 20x 3x cosx 3ex f x 20x 3x cosx 3e O f x 20x 3x COSX 3e x

Find f 0 for the piece wise function f x 4 x 2 if x 2 X 4 if x 2 f 0