Calculus Questions

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

Use the graph to estimate the average rate of change of the percentage of new employees from 1994 to 2000 from 2000 to 2006 and from 1994 to 2006 ICCEI What is the average rate of change of the percentage of new employees from 1994 to 2000 per year Type an integer or a decimal rounded to two decimal places as needed What is the average rate of change of the percentage of new employees from 2000 to 2006 per year Type an integer or a decimal rounded to two decimal places as needed What is the average rate of change of the percentage of new employees from 1994 to 2006 per year Type an integer or a decimal rounded to two decimal places as needed E Percent 30 25 20 151 10 5 0 5 10 Employment Year

Determine whether the function shown is continuous over the interval 6 6 Is the given function continuous over the interval 6 6 Yes 6 4 6 4 y f x 2 4

The graph shows the number of trips taken annually on a certain state s mass transportation system Find the average rate of change of the number of trips from 1992 to 1995 from 1995 to 2000 and from 1992 to 1998 10 9 8 Annual Trips on a State s Mass Transportation System 8 4 million 7 1992 7 5 million 1994 1996 9 6 million 8 9 million 1998 2000 The average rate of change from 1992 to 1995 is Round to three decimal places as needed The average rate of change from 1995 to 2000 is Round to three decimal places as needed The average rate of change from 1992 to 1998 is Rould to three decimal places as needed million per year million per year million per year

Determine whether the function is continuous over the interval 6 6 Is the function continuous over the interval 6 6 6 6 z

he total number of words M t that a person can memorize in t minutes is shown in the graph elow Complete parts a and b to the right AM 40 30 20 10 0 24 28 16 24 8 12 32 28 8 16 24 36 28 32 3640 ote Graph not drawn to scale Q 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 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 The average rate of change of M as t changes from 16 to 24 is words min Simplify your answer 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 vertical and therefore has no slope OB The graph becomes vertical and therefore has an undefined slope OC The graph becomes horizontal and therefore has an undefined slope D The graph becomes horizontal and therefore has a slope of zero

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

a Find a simplified form of the difference quotient and b complete the following table f x 2x f x h f x h b Fill in the table a X X 3 3333 f h 3 2 itt 3 1 3 0 1 0 01 2 1 0 1 0 01 f x h f x h f x h f x h Type int

The initial substitution of x a yields the form x 3x 154 lim x 11 X 121 x 3x 154 x 121 B The limit does not exist Simplify the function algebraically or use a table or graph to determine the limit If necessary state that the limit does not ex Select the correct choice below and if necessary fill in the answer box to complete your choice OA lim X 11 COLLE Type an integer or a simplified fraction

Determine the value for c so that lim f x exists x 3 f x 1 3x c for x 3 C x 6 for x 3 The value of cis

12x in 00 14x sin y dy dx Eetch the region of integration Choose the correct graph below xx X 2x Evaluate the integral 14x sin y dy dx Q Q 5 08 2x x 2x Q oo 9 2x W 2x Q OON Q O

Find the average value of f x y sin x y over a the rectangle 0 x 3 0 y 5 3 b the rectangle 0 x 2 3 0 y 7 6 a The average value of fis 0 Type an exact answer using as needed b The average value of f is Type an exact answer using it as needed

In a particular city taxicabs charge passengers 2 20 for entering a cab and then 0 30 for each one fifth of a mile or fraction thereof traveled If x represents the distance traveled in miles then C x is the cost of the taxi fare where C x 2 20 if x 0 C x 2 50 if 0 x 0 2 and so on The graph of C is shown to the right Use the graph to find each of the following limits if it exists Select the correct choice below and if necessary fill in the answer box to complete your choice A lim C x x 0 2 Simplify your answer Do not include the symbol in your answer OB The limit does not exist Select the correct choice below and if necessary fill in the answer box to complete your choice OA lim C x X 0 2 Simplify your answer Do not include the symbol in your answer B The limit does not exist Select the correct choice below and if necessary fill in the answer box to complete your choice A lim C x X 0 2 Simplify your answer Do not include the symbol in your answer The limit does not exist Cat Cost of taxi in dollars 3 7 3 4 3 1 2 8 2 5 2 24 0 NE 0 4 0 6 0 8 Distance traveled in miles 0 2

In a particular city taxicabs charge passengers 3 20 for entering a cab and then 0 40 for each one fifth of a mile or fraction thereof traveled If x represents the distance traveled in miles then C x is the cost of the taxi fare where C x 3 20 if x 0 C x 3 60 if 0 x 0 2 and so on The graph of C is shown to the right Use the graph to find each of the following limits if it exists Select the correct choice below and if necessary fill in the answer box to complete your choice O A lim C x X 0 26 Simplify your answer Do not include the symbol in your answer B The limit does not exist Select the correct choice below and if necessary fill in the answer box to complete your choice OA lim C x x 0 26 Simplify your answer Do not include the symbol in your answer OB The limit does not exist Select the correct choice below and if necessary fill in the answer box to complete your choice OA lim C x x 0 26 Simplify your answer Do not include the symbol in your answer B The limit does not exist C Cost of taxi in dollars 5 6 5 2 4 8 44 4 3 69 3 24 0 N 0 2 0 4 0 6 0 8 Distance traveled in miles 1

Evaluate the double integral over the given region R R 18y 12x dA R 0 x 4 0sys1

a Write an iterated integral for SS dA over the region bounded by y 7 x y 1 and x 0 using R vertical cross sections b Write an iterated integral for horizontal cross sections OA dx dy SS a Select the correct answer below and fill in the answer boxes to complete your choice OB S dy dx S SS dA over the region bounded by y 7 x y 1 and x 0 using R

Ketch the region bounded by the given lines Then express the region s area as an iterated double integral and evaluate the integral X y x y and y 4 Sketch the region Choose the correct answer below VA ATTR 4 8 12 16 O O OB 16 12 8 43 4 OB The area can be expressed as the double integral OA The area cannot be expressed as one double integral It can be expressed as the sum Express the region s area using one or more iterated double integrals with increasing limits of integration Select the correct choice below and fill in the answer boxes to complete your cho S S dy dx OC The area can be expressed as the double integral dx dy 00 OD The area cannot be expressed as one double integral It can be expressed as the sum S S dx dy SAJ dx dy GEEEE 00 00 OC Q Q 16 12 34 G

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

ketch the region of integration and evaluate the integral TX SS14x 14x sin y dy dx 0 0 A Sketch the region of integration Choose the correct graph below Ay 2 1 0 CAMATER TX SS14x FROM AU YETER R Evaluate the integral 21 X 14x sin y dy dx Q 00 Type an exact answor uning OB AY 2T T X 2 Q Ay 2T T R

4 1 3 0 4 3 Evaluate the iterated integral 20x y dy dx 4 1 4 3 20x y dy dx Simplify your answer 3 0

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

Use Fubini s Theorem to evaluate Apply Fubini s Theorem 3xexy Gellan ng3 RECENTLY Honorer CROCHERISING SS3 00 exy dy dx Tyne exact opp 3xexy dx dy

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

Evaluate the double integral over the given region R S Sex YdA R ex YdA R 0 x In 2 0 ys In 5 Choose two integrals that are equivalent to DA C In 5 In 2 0 S Sex ye R 0 In 5 In 2 S S exy dx dy 0 0 dA ex y dy dx Ssex s R www dA Type an integer or a simplified fraction B In 2 In 5 S S ex Y dy 0 0 In 2 In 5 S 0 0 Sex y dx c

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

The radius of a sphere is increasing at a rate of 5 mm s How fast is the volume increasing in mm s when the diameter is 60 mm Round your answer to two decimal places mm s

The height in meters of a projectile shot vertically upward from a point 3 m above ground level with an initial velocity of 21 5 m s is h 3 21 5t 4 9t after t seconds a Find the velocity in m s after t seconds v t Find the velocity in m s after 2 seconds and after 4 seconds v 2 m s v 4 m s b What is the value in m s of v t when the projectile reaches its maximum height v t m s When in s does the projectile reach its maximum height Round your answer to two decimal places S c What is the maximum height in m Round your answer to two decimal places m d When in s does it hit the ground Round your answer to two decimal places S e With what velocity in m s does it hit the ground Round your answer to two decimal places m s

Use logarithmic differentiation to find the derivative of the function y x6 sin x y x

Use the graph to state the absolute and local maximum and minimum values of the function Assume each point lies on the gridlines Enter your answers as a comma separated list If an answer do not exist enter DNE y 5 4 3 2 1 1 absolute maximum value absolute minimum value local maximum value s local minimum value s A y f x 2 3 4 5 6 e

Finally setting f c 4c 4 equal to 0 and solving for c gives the following result Enter your answers as a comma separated list 4c 4 0 C

Since we have verified that all three hypotheses of Rolle s theorem are satisfied we know that there is a number c in 1 3 such that f c 0 To find this number c we begin by finding f x f x 2x 4x 9 f x Therefore we have the following f c

y g x W 1 C 0 g is shown 1 i a Which of the following verifies that g satisfies the hypotheses of the Mean Value Theorem on the interval 0 8 Select all that apply Og is continuous on the closed interval 0 8 Og obtains at least one maximum on the closed interval 0 8 g is differentiable on the open interval 0 8 Og obtains at least one minimum on the open interval 0 8 Og is continuous on the open interval 0 8 Og takes only positive values on the closed interval 0 8 b Estimate the value s of c that satisfy the conclusion of the Mean Value Theorem on the interval 0 8 Enter your answers as a comma separated list Round your answers to one decima places If an answer does not exist enter DNE c Estimate the value s of c that satisfy the conclusion of the Mean Value Theorem on the interval 2 6 Enter your answers as a comma separated list Round your answers to one decimal places If an answer does not exist enter DNE

The third hypothesis is satisfied if f a f b To verify that the given function satisfies this hypothesis on the interval 1 3 we must evaluate f 1 and f 3 and compare the re f x f 1 f 3 Since these values Select the same the third hypothesis Select satisfied 2x 4x 9

Tutorial Exercise Verify that the function satisfies the three hypotheses of Rolle s theorem on the given interval Then find all numbers c that satisfy the conclusion of Rolle s theorem f x 2x 4x 9 1 3

Consider the following function g v v 48v 8 Find the derivative of the function g v Find the values of v such that g v 0 Enter your answers as a comma separated list If an answer does not exist enter DNE V Find the values of v in the domain of g such that g v does not exist Enter your answers as a comma separated list If an answer does not exist enter DNE V Find the critical numbers of the function Enter your answers as a comma separated list If an answer does not exist enter DNE V

Sketch the graph of f and use your sketch to find the absolute and local maximum and minimum values of f Enter your answers as a comma separated list If an answer does not exist enter f x 2 3x x 1 absolute maximum value absolute minimum value local maximum value s local minimum value s

Explain the difference between an absolute minimum and a local minimum O A function f has an absolute minimum at x c if f c is the smallest function value on the entire domain of f whereas f has a local minimum at c if f c is the smallest function value when x is near c O A function f has an absolute minimum at x c if f c is the smallest function value when x is near c whereas f has a local minimum at c if f c is the smallest function value on the entire domain of f O There is no difference O A function f has an absolute minimum at x c if f c is the largest function value on the entire domain of f whereas f has a local minimum at c if f c is the largest function value when x is near c O A function f has an absolute minimum at x c if f c is the largest function value when x is near c whereas f has a local minimum at c if f c is the largest function value on the entire domain of f

For each of the numbers a b c d r and s state whether the function whose graph is shown has an absolute maximum or minimum a local maximum or minimum or neither a maximum nor minimum Enter your answers as a comma separated list y absolute maximum local maximum M absolute minimum local minimum a b C d r

y x y x6x X Enhanced Feedback y Third multiply y Please try again First take the logarithm on both sides to get an expression for In y Second differentiate both sides with respect to x to obtain an expression for the expression of y to finally get the expression for y In most cases when logarithmic differentiation

If f x 4 sin x In 3x find f x f x Graph fand f in the same viewing window y 8 6 4 2 O 2 4 W X 47 2x 2 f 2 6 2x X 4x P

A cylindrical tank with radius 3 m is being filled with water at a rate of 2 m min How fast is the height of the water increasing in m min m min

Suppose f and g are differentiable functions and let h x f x 9 x then h x is given by the following form h x g x f x 91 f x In f x f x 9 g x Use this formula to find the derivative for each of the following functions a h x x5 h x b h x 5x h x c h x cos x h x

Let V represent the volume of a sphere with radius r mm Write an equation for V in mm in terms of r V r mm Find the radius of a sphere in mm when its diameter is 60 mm r mm The radius of a sphere is increasing at a rate of 5 mm s How fast is the volume increasing in mm s when the diameter is 60 mm Round your answer to two decimal p mm s

The radius of a sphere is increasing at a rate of 4 mm s How fast is the volume increasing in mm s when the diameter is 60 mm Round your answer to two decimal places m s mm

a If A is the area of a circle with radius r and the circle expands as time passes find dA dt dA dt dr dt in terms of dr dt b Suppose oil spills from a ruptured tanker and spreads in a circular pattern If the radius of the oil spill increases at a constant rate of 2 m s exactly how fast in m s is the area of th increasing when the radius is 29 m m s

Each side of a square is increasing at a rate of 7 cm s At what rate in cm s is the area of the square increasing when the area of the square is 9 cm cm s

Differentiate the function y y In 3 8x 3 8x

Find the derivative of the function f x sin 8x f x X