Application of derivatives Questions and Answers

4 2 X 2 2 X y f x 6 8 a On what interval s is f increasing Enter your answer in interval notation x On what interval s is f decreasing Enter your answer in interval notation b At what value s of x does f have a local maximum Enter your answers as a comma separated list At what value s of x does f have a local minimum Enter your answers as a comma separated list c On what interval s is f concave upward Enter your answer in interval notation

Write the equation log 216 x in exponential form

A certain biological compund is known to have exponential growth The initial measurement show there is 3 grams of the compound One day later there is 3 7 grams of th compound Thinking of the exponential equation A Aek find the value of k kt O a k 0 1769 Ob k 1 1935 Ock 0 2097 d k 1 1564

Tutorial Exercise Find the limit Use l Hospital s Rule where appropriate If there is a more elementary method consider using it lim X 0 cos x 1 x 4x4

5 4 3 2 1 12 3 4 5 6 x a Find the interval s on which f is increasing Enter your answer using interval notation b Find the interval s on which f is decreasing Enter your answer using interval notation c Find the interval s on which fis concave upward Enter your answer using interval notation d Find the interval s on which fis concave downward Enter your answer using interval notation

For each of the following state the aysmptote Be sure to state it as the equation of a line a y 2 2 b y in x 2 C y In x 1 d y 2x 2

The logistic growth function describes the number of people in a particular community who believe an internet conspiracy days after its initial post 12 000 y 1 199e 0 625t a How many people believed shared the conspiracy initially b How many people believed in the conspiracy after the 4th day c What is the carrying capacity and what does it mean

Angle has a measure of of 32 Provide your answer below What is its measure in degrees

Convert the angle 200 into radians Enter an exact fraction that contains the symbol

What is the radian measure of 210 Enter an exact fraction that contains the symbol

3 A bee is flying around and its velocity function is given by v t 1 e 3t and has initial position at the point 2 1 0 Where does the bee hit the surface of th parabolic cylinder z x 9 and what is the speed at which the bee hits this surface Us technology to help approximate the solutions

The total number of words M t that a person can memorize in t minutes is shown in the graph below Complete parts a and b to the right 40 AM 30 20 10 0 24 26 16 22 8 10 32 26 8 16 24 36 26 323640 Note Graph not drawn to scale t Q 5 a Find the average rate of change of M as t changes from 0 to 8 from 8 to 16 from 16 to 24 from 24 to 32 from 32 to 36 The average rate of change of M as t changes from 0 to 8 is Simplify your answer The average rate of change of M as t changes from 8 to 16 is words min Simplify your answer The average rate of change of M as t changes from 16 to 24 is Simplify your answer words min The average rate of change of M as t changes from 24 to 32 is Simplify your answer words min words min words min The average rate of change of M as t changes from 32 to 36 is Simplify your answer b Explain the behavior of the average rates of change after 24 min OA The graph becomes horizontal and therefore has an undefined slope OB The graph becomes vertical and therefore has no slope OC The graph becomes horizontal and therefore has a slope of zero OD The graph becomes vertical and therefore has an undefined slope

We can now use the substitution u sin 8 so du 8 sec 4 u Step 4 sec sin 8 Jn 3 15 4 We have determined that if we let u sin 8 then du cos 8 de on the interval 4 cos 8 de sin 0 du 15 4 5 2 Step 5 We can now evaluate the integral 15 4 5 2 1 15 4 U 2 cos 8 15 4 24 cos 8 du de Once more we must also make a substitution for the limits of integration When 8 TT 15 VS VTS Applying the substitution gives us the following result

Find the average value of f x 00 x 1 X 1 x 3

Consider the following dt Let t 9 tan 8 and determine the following a r9 1 dt V 81 t t 0 8 sec 0 9 sec t 9 8 X 0 T 4 Evaluate the integral 2 1 X de

Question 5 Find an equation of the circle that has center 5 1 and passes through 1 5 0

Consider the following y 7x In x 1 7 Find an equation of the tangent line to the graph of the function at the given point Use a graphing utility to graph function and its tangent line at the point and use the tangent feature of a graphing utility to confirm your results y

In a statistics class there are 23 juniors and 19 seniors 9 of the seniors are females and 15 of the juniors are males A student is selected at random Find the probability of selecting a junior or a female 32 42 28 42 O 24 42 34 42 O 30 42

Two dice are rolled one time Find the probability of getting a sum of 6 or doubles O 10 36 11 36 O 12 36 9 36 13 36

A couple has three children Find the probability of getting two boys 18 48 0 3 8 05 8

Two coins are tossed one time Find the probability of getting one head O 2 1 8 4

There are 10 members on a board of directors If they must elect a chairperson a secretary and a treasurer how many different slates of candidates are possible Write the answer as a whole number

A symphony decides to raise money by raffling off an oven worth 500 a dinner worth 170 and a set of books worth 125 A total of 2000 tickets are sold at 2 each After each prize drawing the winning ticket is returned to the pool of tickets Find the expected value of winning for a person who buys 1 ticket in the raffle 1 38 O 1 25 O 1 75 O 1 95 O 1 60

A bag contains 5 cherry 2 orange and 3 lemon candies You reach in and take 3 pieces of candy at random Find the probability of selecting 1 orange and 2 cherry 0 385 0 274 0 047 0 495 0167

Maximize Subject to Maximum is X y z 4x 3y 3x 7y 5x 3y X Y 56 50 0 0

Use the graph of the derivative f of a continuous function f is shown Assume f continues to co y 4 X 2 2 X 2 y f x a On what interval s is f increasing Enter your answer using interval notation 4 On what interval s is f decreasing Enter your answer using interval notation b At what value s of x does f have a local maximum Enter your answers as a comma separated list At what value s of x does f have a local minimum Enter your answers as a comma separated list c On what interval s is f concave upward Enter your answer using interval notation On what interval s is concave downward Enter your answer using interval notation

Find the critical numbers of the function Enter your answers as a comma separated list If an answer does not exist enter DNE 1 2 2 f x 9 1 X 5

C Problem 70 Summarize the pertinent information as in previous problem obtained by applying the graphing strategy and sketch the graph of f x 2e0 5x e 0 5x

4 T shirt manufacturer is planning to expand its workforce It estimates that the number of T shirts produced by hiring x new workers is given by T x 0 25x 3x 0 x 9 When is the rate of change of T shirt production increasing and when is it decreasing What is the point of diminishing returns and the maximum rate of change for T shirt production Graph T and T on the same coordinate system The rate of change of T shirt production is increasing on Type your answer in interval notation

C Problem 36 Find the intervals on which the graph of F is concave upward the interv which the graph of f is concave downward and the x y coordinates of the inflection points f x x 2x 5x 3

Problem 46 Find the intervals on which f x is increasing the intervals on which f x is de creasing and the local extrema f x x 2 e

Find the average value of f x IN 2 X 1 x 3

Evaluate the integral 6 0 75 2 9 16x dx

Consider the following function 3 dx a Determine an appropriate trigonometric substitution O Use x 8 sin 0 where O Use x 8 tan 0 where 1 IT 2 IT 2 since the integrand contains the expression 8 2 8 TT 2 since the integrand contains the expression 82 x O Use x 8 sec 0 where 0 0 or 0 3 2 since the integrand contains the expression 8 x 2 b Apply the substitution to transform the integral into a trigonometric integral Do not evaluate the integral x3 J dx 64 x de

Consider the following given function and given interval g x x 2 0 2 a Find the average value gave C b Find c in the given interval such that gave g c Enter your answers as a comma separated list Round your answers to three decimal places 60 c Sketch the graph of g and a rectangle whose base is the given interval and whose area is the same as the area under the graph of g y y y 40 20 gave O of of g on the given interval 0 5 1 0 1 5 2 0 20 15 10 5 O S 0 5 1 0 1 5 2 0 20 15 10 S os 0 5 1 0 1 5 2 0 X 60 40 20

If 5000 is invested at 7 annual simple interest how long does it take to be worth 8150 yr

FIne the area enclosed by the given ellipse x asint y bcost 0 t 2pi The area is

Graph the given function and then find the specified limits If necessary state that the limit does not exist g x x 9 lim g x and lim g x X 3 x 1 Choose the correct graph below OA O C OB O D Ay Q Q

Find an equation of the line tangent to the graph of f x 5 6x at 5 145 The equation of the tangent line to the graph of f x 5 6x at 5 145 is y Type an expression using x as the variable

Use the graph to find the following limits and function value a lim f x X 2 b lim f x X 2 c lim f x X 2 d f 2 O A O A lim f x Type an integer X 2 a Find the limit Select the correct choice below and fill in any answer boxes in your choice OA lim f x Type an integer X 2 2 OB The limit does not exist b Find the limit Select the correct choice below and fill in any answer boxes in your choice lim f x Type an integer X 2 2 2 OA 1 2 X OB The limit does not exist c Find the limit Select the correct choice below and fill in any answer boxes in your choice Type an integer B The answer is undefined 57 OB The limit does not exist d Find the function value Select the correct choice below and fill in any answer boxes in your choic

Find g t for the function g t m 3

Find H x for the given function H x 3 x 8 H x Type an exact answer using radicals as needed

K By determining f x lim h 0 f x 7x f x h f x h f 1 Simplify your answer find f 1 for the given function

K An object dropped from rest from the top of a tall building on Planet X falls a distance d t 19t feet in the first t seconds Find the average rate of change of distance with respect to time as t changes from t 2 to t 8 This rate is known as the average velocity or speed The average velocity as t changes from 2 to 8 seconds is Simplify your answer feet sec

e integrals below give the area of a region in the xy plane Sketch the region label each bounding curve with its equation and gi e coordinates of the points where the curves intersect Then find the area of the region x COS X S S sinx dy dx Sketch the region label each bounding curve with its equation and give the coordinates of the points where the curves intersect Choose the correct answer below y cosx sinx The tou Q B NI The area of the region is Type an exact answer using radicals as needed y cos x y sinx X K 1st Q Q G KIN OC y sinx y cos x de fre Q

3 70 0 4x A ketch the region of integration Choose the correct graph below 3 dy dx 20 Evaluate the integral x OB Ay 20 20 Q dy dx Type an integer or a simplified fraction 5 Ay 20 20 Q O Q

valuate the iterated integral 5 0 0 2 x y 8xy dy dx 50 S S x y 8xy dy dx 0 2 Type an exact answer in sim

Show that the equation x3 14x c 0 has at most one solution in the interval 2 2 Let f x x 14x c for x in 2 2 If f has two real solutions a and b in 2 2 with a b then Select implies that there is a number r in a b such that f r 0 Select Now f r equation cannot have two real solutions in 2 2 Hence it has at most one real solution in 2 2 Since the polynomial f is continuous on a b and differentiable on a b Since r is in a b which is contained in 2 2 we have r 2 so r2 4 It follows that 3r2 1473 4 14 2 0 This contradicts f r 0 so the gi

Identify any vertical horizontal or oblique asymptotes in the graph of y f x State the domain of f A There is one vertical asymptote Its equation is Type an equation Identify any vertical asymptotes Select the correct choice below and if necessary fill in the answer box to complete your choice B There are two vertical asymptotes The equation of the leftmost one is the equation of the rightmost one is Type equations OC There are no vertical asymptotes 20 Identify the horizontal or oblique asymptote if one exists Select the correct choice below and if necessary fill in the answer box to complete your choice OA The equation of the horizontal asymptote is Type an equation B The equation of the oblique asymptote is Type an equation in slope intercept form OC There is no horizontal asymptote The domain is Q and

A plane flying horizontally at an altitude of 1 mile and a speed of 550 mi h passes directly over a radar station Find the rate at which the distance from the plane to the station is increasing when it has a total distance of 5 miles away from the station Round your answer to the nearest whole number mi h