Calculus Questions

Find the arc length of the curve below on the given interval X 8y 2 for 3 sys5 The length of the curve is Simplify your answer

Find a curve that passes through the point 1 7 and has an arc length on the interval 2 6 given by 2 What is one such curve y 12 2 5 S 6 1 576x dx

60 Find tan if is the angle formed by the line y mx and the positive x axis B 1 m

Find the area of the region enclosed by the curves y 2 sinx and y 2 sin 2x for 0sxsx The area of the region enclosed by the curves is Type an exact answer using radicals as needed y 2 sinx ly 2 sin 2x

a Write and simplify the integral that gives the arc length of the following curve on the given interval b If necessary use technology to evaluate or approximate the integral T 2 y 2 sin x for SXS 3 3 a Set up the integral that gives the arc length of the curve Select the correct choice below and fill in the answer box to complete your choice Type an exact answer OA L OB L 2 SO dy 2 2x JOdx

59 Find the slope of the line y mx It passes through the origin and the point 1 m

6 Find a curve that passes through the point 1 7 and has an arc length on the interval 2 6 given by 1 1 576x d 2 What is one such curve

What differentiable functions have an arc length on the interval a b given by the following integrals Note that the answers are not unique Give all functions that satisfy the conditions a b 1 64x dx a E y 8x C G a Choose the correct answer below Select all that apply A y 8x C C y 32x C y 4x6 15 b b S a C S T 1 64 cos 2x dx B D F OH y 32x C y y 8x5 5 8x5 y C 5 C 4x6 15 C

7 The line y 3x passes through the points 0 0 and 1 3 Find its slope

8 Suppose the angle formed by the line y 3x and the positive x axis is 8 Find the tangent of 0 y y 3x 1 3 0

The function defined by years and x 0 corresponds to 1960 325 b When do the seasonal maximum and minimum carbon dioxide levels occur 3 7 11 2 2 2 OA Maximums x 0 OB Maximums X 15 2 2 3 7 11 2 22 159 OC Maximums X 51515 2 2 2 OD Maximums X Minimums x 222 Minimums x 0 Minimums x 5 www Minimums x 2 2 2 2 2 159 325

Is the graphed function differentiable at x 3 Choose the correct answer below OA Yes because f x has a non vertical tangent line at x 3 OB No because the tangent line to f x at x 3 is vertical OC Yes because f x has two tangent lines at x 3 both non vertical Next question C 10 8 6 4 2

53 cos 0 csc 0 5 and 8 E QII

Suppose the graph of f on the interval a b has length L where f is continuous on a b Evaluate the following integrals in terms of L b 2 1 2x dx a 2 a b b c S f x dx ifc 0 a c

53 cos 8 csc 8 5 and 8 E Q

The graph to the right shows the total sales in thousands of dollars for a department store during a typical 24 hour period a Calculate the rate of sales during the period between 8 A M and 10 A M b Which 2 hour interval in the day sees the highest rate of sales and what is the rate Sales in Thousands of Dollars 24 20 16 12 8 0 2 A M 6 A M 12 noon 6 P M 10

40 Suppose f is an odd function and is differentiable everywhere Show that fo every positive number 6 there exists a number cin b b such that f b f c b

1 12 13 and 14 Verify that the function satisfies the three hypotheses of Rolle s heorem on the given interval Find all numbers c that satisfy the conclusion of olle s Theorem 11 f x 2x 4x 5 1 3 12 f x x 2x 4x 2 2 2

24 Let f x 2 2x 11 Show that there is no value c in 0 3 such that f 3 f 0 f c 3 0 Explain why this does not contradict the Mean Value Theorem

The owner of a photocopy store charges 8 cents per copy for the first 100 copies and 6 cents per copy for each copy exceeding 100 In addition there is a setup fee of 2 30 for each photocopying job a Determine R x the revenue from doing one photocopying job consisting of x copies b If it costs the store owner 3 cents per copy what is the profit from doing one photocopying job consisting of x copies Recall that profit is revenue minus cost a Choose the correct answer below OA O C R x R x 08x 2 30 1 x 100 14x 2 30 100 x 08x 2 30 1 x 100 100 x 06x OB O D R x R x 08x 2 30 1 x 100 06x 4 30 100 x 08x 2 30 1 x 100 06x 2 30 100 x

Is the graphed function differentiable at x 1 Choose the correct answer below OA Yes the function is differentiable at x 1 OB No because while the function is continuous at x 1 it does not have a tangent line at x 1 OC No because the function is not defined at x 1 OD No because while the function is defined at x 1 and lim exists lim is not equal to the value of the function at x 1 X 1 X 1 OE No because while the function is continuous at x 1 it has a vertical tangent line at x 1 OF No because while the function is defined at x 1 lim does not exist X 1 10 8 r 4 2 6 y 2

17 18 19 and 20 Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval Find all numbers c that satisfy the conclusion of the Mean Value Theorem 17 f x 2x 3x 1 0 2 18 f x x 3x 2 2 2 19 f x lnx 1 4 1 20 f x 1 3

The given function is defined for all x except for one value of x If possible define f x at the exceptional point in a way that makes f x continuous for all 8 x 64 f x X 0 Select the correct choice below and if necessary fill in the answer box to complete your choice OA f x 8 x 64 X for x 0 for x 0 OB There is no range value that will make f x continuous at x 0

Determine if the function f x is continuous and or differentiable at x 3 f x x 18 for 3 x 3 7x for 3 x 8 Is f x continuous at x 3 O Yes No

Determine the value of a that makes the function f x continuous at x 9 x 45 x 9 x29 f x ax Select the correct choice below and if necessary fill in the answer box to complete your choice O A a

Determine whether the following function is continuous and or differentiable at x 10 f x x Is the function continuous at x 10 OA Yes the function is continuous at x 10 OB No because while the function is defined at x 10 and lim f x exists lim f x is not equal to the value of the function at x X 10 X 10 OC No because the function is not defined at x 10 OD No because while the function is defined at x 10 lim f x does not exist X 10

Determine whether the following function is continuous and or differentiable at x f x Is the function continuous at x 6 OYes 2x 6 for 0 x 6 6 for 6 x No

4 Ex 3 Graph f x x y Name the asymptotes Domain Range X X Y

Is the graphed function differentiable at x 5 Choose the correct answer below OA Yes the function is differentiable at x 5 OB No because while the function is continuous at x 5 it has a vertical tangent line at x 5 OC No because while the function is continuous at x 5 it does not have a tangent line at x 5 OD No because the function is not defined at x 5 OE No because while the function is defined at x lim does not exist X 5 OE No because while the function is defined at x 5 and lim exists lim is not equal to the value of the function at x 5

Is the function graphed to the right continuous at the point x 1 Choose the correct answer below 54 3 2 1

The graph shows an idealized description of the temperature in F for approximately the last 150 thousand years of a location at the same latitude Complete parts a c www a Find the highest and lowest temperatures recorded The highest temperature is The lowest temperature is F F Average Annual Temperature Idealized 140 000 105 000 70 000 Years ago 80 70 60 150 40 30 35 000 10 F

Ex 2 Graph f x x y Name the asymptotes X Domain leftmost to rightmost Range lowest to highest X Y

function graphed is of the form ya sin bx or ya cos bx where b 0 Determine equation of the graph 1 5 3 1 1 3 A

The function graphed is of the form y a sin bx or y a cos bx where b 0 Determine the equation of the graph y Use integers or fractions for any numbers in the expression 6 Ay 6

Determine the amplitude and period of the given function Graph the function 2 X y 3 sin The amplitude is Simplify your answer Type an exact answer using it as needed

Use the given graph to estimate the value of each derivative a f 3 b 1 2 c f 1 d f 0 e f 1 1 f 2 9 f 3 y f x Sketch the graph of f 2 7 N ya 3 y 3 2 3 4 3

Find the required trigonometric ratio of using the given information 45 tan 0 sin 0 and 8 E QI and 6 and 0 E QI 2 3 46 tan 0 cos 0 3 4

Write and simplify but do not evaluate an integral with respect to x that gives the length of the following curve on the given interval y 5 cos 3x on TR 2 4 An integral that gives the arc length is Tuno exact answers 4 I RIN 5 cos 3x dx

Use the figure below which shows the graph of y f x to answer the following questions 1 0 A Between x 3 and x 3 B Between x 3 and x 0 Click on the graph to ge A Estimate the integral f f x dx You will certainly want to use an enlarged version of the graph to obtain your estimate B Which of the following average values of f is larger 4 X

Find the arc length of the line y 4x 3 on 0 3 using calculus and verify your answer using geometry Which of the following is a simplified form of the integral that gives the arc length of the described line segment 3 OA S 1 4x 0x 4x d dx B S 17 dx c S 3 5 dx OD 1 16x dx 0 The length of the curve is

Write and simplify but do not evaluate an integral with respect to x that gives the length of the following curve on the given intervall y 5 cos 3x on SIRET TT 2 4 An integral that gives the arc length is dx COLDS

is the average value of f x on 0 x 2 alue is the average value of g x on 0 x 2 alue is the average value of f x g x on 0 x 2 alue e following statement true 0 5 0 5 0 5 0 5 f x 015 2 0 5 0 5 F 8 5 1 1 g x 1 1

Find the remaining trigonometric ratios of 12 21 cos 0 and terminates in QI 22 13 23 cos 0 25 sin 0 5 13 and 0 E QIII 27 sin 0 and 0 QIV E and terminates in QI 29 tan 0 and 0 terminates in QI 4 31 tan 0 3 and 8 E QII 33 sec 0 2 and 0 terminates in QII 35 sec 0 3 and 8 E QIII 5 37 csc and terminates in QII 0 39 csc 0 1 13 5 and 0 QIII QU 24 26 28 30 32 34 s 36 S 38 c 40 c

Among all rectangles that have a perimeter of 64 find the dimensions of the one whose area is largest Write your answers as fractions reduced to lowest terms

19 If sin 0 20 If cos 0 23 cos 25 sin 1 V5 27 sin 0 Find the remaining trigonometric ratios 12 21 cos 0 and terminates in QI 13 5 13 and 8 E QII find cos 1 1 V10 2 and E QIII fine and terminates in QI a and 8 E QIII a and 8 E QIV 3 29 tan 0 and 0 terminates in Ol

Find the remaining trigonometric ratios of 12 and terminates in QI 22 13 21 cos 0

9 sin 5 13 and cos

17 If cos 0 V3 2 and terminates in QI find sin 0

13 Find cose if sin and terminates in QII

5 3 Points DETAILS SCALCET9 2 XP 8 004 Trace or copy the graph of the given function f Assume that the axes have equal scales Use estimates of the derivative of f to sketch the graph of f on the same coordinate axes y O Q 20 DO