Application of derivatives Questions and Answers

3 n n is an n th Riemann sum on the interval 0 1 so AX Rn n lim Rn n x n n HINT The limit is a definite integral from 0 to 1 You just have to figure out the function being integrated A 1 R 1 2 C 3 2 D 2 3 E 2

20 Which of the following statements about indefinite integrals are true I f f x g x dx f f x dx f g x dx II f f x g x dx f f x dx f g x dr III ff g x g x dx f g x C f x 1 IV f f x dx n 1 C A only I and II are true B only I and III are true C only E only I I and IV are true D only I II and IV are true III and IV are true

17 1 4x dx A VI 4x C B 1 4r 3 2 C LIN C 1 4x 3 2 C D x 1 4x C 1 VIE E 1 4x C

5 g x x tan x sin x g x A 2x sec x cos x B 2x tan x sin x x 1 1 x 1 1 x C r 3 tan x sin x x sec r cos x D 2x tan x sin x x sec x cos x

9 If h x x 4 sin x 2 then h x anonton A 2x cos x 2 B 4x2 sin x 2 C 4x cos x 2 D 2x sin x 2 x 4 cos x 2

2 lim sin 4x sin 6x x 0 uds 0 C 0 D 1 E Does not exist A 3 2 B 2 3 C 0

8 If y A x 1 x 1 D then dy OT du dx x 3x 2x 2xx x 1 2x 3x x B x 1 3x2 pex 1 G o x 1 2x 3x x 1 da nat x 1 C ont

7x4 8x 13x4 16x 1 lim a x 5 A B 0 C 1 D 1 2 E 13

w x cos x 1 sin x 0 T

x 7 7 x x y

c x x 1 1 3

s x 3e2ln x 1 4

2 For the following functions find the critical points determine whether each critical point is a local minimum local maximum or neither and provide the global minimum and global maximum on the specified interval 1 2 3 Function f x x 4x 6x 2 g x h x eln x 1 x 3x x 1 Interval 3 5 0

T 13 14 b x 2 24 2x3 5x2 1 x 2 c x x 1 4 4 1 3

14 Given the graph for f has a domain of 2 6 and range of 0 8 Let g x 3f x 2 Find the domain and range of g x

7 Use the graph of f to find the following J a Domain 0 3 3 47 0 4 00 b Zero s 2 points each c Limit Notation of Vertical Asymptote s d End Behavior

13 cos 0 120

2 2 2 d

8 4 points The number of bacteria in a culture is increasing according to the law of exponential growth n t noe L of bacterial after 2 5 hours Initially there are 100 bacteria and after 1 hours there are 400 bacteria Find a formula for the number n t nert

Given the curve x xy 2y 6 a Find an expression for the slope of the curve at any point x y on the curve b Write an equation for the line tangent to the curve at the point 2 1 c Find the coordinates of all other points on this curve with slope equal to the ope at 2 1

3 Given the curve x xy y 9 a Write a general expression for the slope of the curve

A person wants to purchase both parcels of land shown in the figure to the right and make a check on their combined area There is no equation for the river frontage so the person uses the average of the left sum of rectangles covering the area The 1 000 foot baseline is divided into 10 equal parts At the end of each subinterval a measurement is made from the baseline to the river and the results are tabulated Let x be the distance from the left end of the baseline and let h x be the distance from the baseline to the river at x Use L 0 to estimate the combined area of both parcels and calculate an error bound for this estimate How many subdivisions of the baseline would be required so that the error incurred in using L would not exceed 2 500 square feet X 0 100 200 300 400 500 600 700 800 900 1 000 h x 0 196 244 254 304 344 413 474 274 527 550 River ft2 Parcel 1 1 000 ft The combined area of both parcels is The error bound for the estimate is ft The number of subdivisions required so that the error incurred in using L would not exceed 2 500 square feet isn Parcel 2 h x 550 ft

8 tan 0 330

7 sin 0 300

300

cos 609

0 cos 0 120

Find the exact length of the polar curve 82 0 0 3 t

60 y sin 0 90 180 270 360 450 5405

2 u 45 V

0 10 60 y

45 y 60 X 7 3 2

48 y 0 X 30 8 3

25 cos 1

24 sin 1

31 tan 3 3

32 sin 2 2

29 cos 1

Find the exact value of each expression 19 cos 1 21 sin 2 nia 8 20 cos 0 38 Flod m 22 sin 0

9 y 6 30

43 sin 0

31 X 60 3 2 y

5 X 30 2

37 cos 2

35 sin 3 2

The population of a city is expected to grow at the rate of x x 4 thousand people per year after x years Find the total change in population from year 0 to year 21 thousand people

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function 1 du Hint 7 52 g x g x u du u du f 52 7x f u du f

The function f x 7x 9x has one local minimum and one local maximum Algebraically use the derivative to answer the questions Leave answers in 4 decimal places when appropriate this function has a local maximum at with value and a local minimum at x with value

Find the sum 31 16 10 2 k 0 X

The logistic growth function f t OA 353 butterflies OB 253 butterflies OC 5 040 butterflies OD 153 butterflies 360 1 8e 0 211 describes the population of a species of butterflies t months after they are introduced to a non threatening habitat How many butterflies are expected in the habitat after 14 months Round to nearest whole numb