Calculus Questions

A ladder 20 ft long rests against a vertical wall Let be the angle between the top of the ladder and the wall and let x be the distance from the bottom of the ladder to the wall If the bottom of the ladder slides away from the wall how fast in ft rad doe x change with respect to when 8 ft rad 2 3

f x COS X f x sin x cot 4 d cot x cos ec x Enhanced Feedback Please try again using the rules for differentiating trigonometric functions Keep in mind that sin x cos x and dx cot x csc x

A mass on a spring vibrates horizontally on a smooth level surface as shown in the figure Its equation of motion is x c 8 sin t where t is measured in seconds and x In centimeters Equilibrium position t x a Find the velocity and acceleration at time t v t X In what direction is the mass moving at that time O b Find the position velocity and acceleration of the mass at time t 2 3

A graphing calculator is recommended Let f x 4e cos x a Find f x and f x f x b Check to see that your answers to part a are reasonable by graphing f f and f y y y y 150 125 100 75 50 25 R N K K X x 7 2 25 50 75 100 100 100 100 125 125 125 125 150 150 DO 150 KIN Describe the relationship between the three graphs Note that f 0 where f has a relative minimum or maximum and f 0 where f has a relative minimum or maximum Also note that f is negative when fis decreasing and f is negative when f is decreasing O Note that f 0 where f has a relative minimum or maximum and f 0 where f has a relative minimum or maximum Also note that f is negative when fis decreasing and f is negative when f is decreasing O Note that f 0 where f has a minimum and f 0 where f has a minimum Also note that f is positive when fis decreasing and f is positive when f is decreasing Note that f 0 where f has a relative minimum or maximum and f 0 where f has a relative minimum or maximum Also note that f is negative when fis decreasing and f is positive when f is decreasing Note that f 0 where f has a maximum and f 0 where f has a maximum Also note that f is positive when f is decreasin

13 1 0 12 Points Find constants A and B such that the function y A sin x B cos x satisfies the differential equation y y 5y sin x A B Need Help Read It Watch It

Need Help ASK YOUR TEACHER DETAILS SCALCET9 3 3 043 MI 9 0 12 Points A ladder 20 ft long rests against a vertical wall Let be the angle between the top of the ladder and the wall and let x be the distance from the bottom of the ladder to the wall If the bottom of the ladder slides away from the wall how fast in ft rad does x cha respect to when 8 Submit Anserer D 0 12 Points Find the limit ft rad Read It DETAILS sin 5x lim X10 7x Watch t Master it MY NOTES SCALCET9 3 3 045 MY NOTES PRACTICE AN ASK YOUR TEACHER PRACTICE A

5 0 12 Points 7 3 Suppose f a g b h K M 3 F 3 4 and f 7 Let g x f x sin x and h x cos x Find the following f x

Let the function f be defined by tan x 1 sec x a Use the Quotient Rule to differentiate the function f x f x f x b Simplify the expression for f x by writing it in terms of sin x and cos x and then find f x f x c Show that your answers to parts a and b are equivalent Yes O No Need Help Submit Answer 10 13 Deinal Read It Watch It

Consider the following figure Find lim S 8 0 d Need Help Submit Answer Disc There is a circular arc of length s and a chord of length d both subtended by a central angle 8 18 0 17 Points d 0 S Read It Watch It DETAILS SCALCET9 3 XP 3 002

Use any method to evaluate the integral 30 dx S 225x 1 30 dx 225x 1 S 0

Evaluate the integral 169 t dt 169 1 dt Type an exact answer using radicals as needed

Use an appropriate substitution and then a trigonometric substitution to evaluate the integral In 3 S 0 In 3 S 0 e dt 2t 4 e dt 2t ve 4 11

Evaluate the integral 2 81 y 81 2 dy y 9

Use any method to evaluate the following integral 2 S 0 2 S 3dx 3 16 x 3dx 3 2 II 16 x Type an exact answer using radicals as needed

Evaluate the integral 16 12 dt 16 F dt dt Type an exact answer using radicals as needed

Use any method to evaluate the integral x dx 3 x dx 3 2 2 x 49 X 49

Evaluate the following integral 7 3 2 S 0 49 x2 7 3 2 S 0 T dx dx 49 x

Evaluate 8 3 3 S 8 3 3 dx 64 x 8 3 3 dx 2 64 x 8 3 3 Type an exact answer using and radicals as ne

Use any method to evaluate the integral x dx 3 S x 2 S x 16 x dx X x 16 2

fis the function defined by f x A B C lim x 2 x 2 z 2 lim x 2 lim 2 x 2 2 lim z 2 2 4 24 2 2 then limf x is equivalent to which of the following z 2

Use the graph of the function to answer parts a through f P x x 5 1 IN The coordinates of the vertex are 5 1 Type an ordered pair b Give the domain and range What is the domain of P x 0 00 Type your answer in interval notation What is the range of P x 1 00 Type your answer in interval notation c Give the equation of the axis of symmetry x 5 Type an equation d Give the largest open interval over which the function is increasing Type your answer in interval notation 10 B

K Use the graph of the function to answer parts a through f P x 4 x 5 1 a Give the coordinates of the vertex The coordinates of the vertex are 5 1 Type an ordered pair b Give the domain and range What is the domain of P x 0 00 Type your answer in interval notation What is the range of P x 0 1 Type your answer in interval notation c Give the equation of the axis of symmetry Type an equation 10 12

Answer the following True False quiz Enter T or F 1 1 g 1 x g x 2 If f c 0 then f has either a local maximum or a local minimum at c 3 Continuous functions are differentiable 4 If f c 0 and f c 0 then f x has a local minimum at c 5 A continuous function on a closed interval a b attains a maximum and a minimum value in a b 6 If f x and g x are increasing on an interval I then f x g r is increasing on I 7 If f x e then f x 2e

f x 9 5e A List all critical numbers of f If there are no critical numbers enter NONE Critical numbers B Use interval notation to indicate where f x is increasing Increasing C List the x coordinates of all local maxima of f If there are no local maxima enter NONE I values of local maxima D Find the x coordinates of all local minima of f If there are no local minima enter NONE I values of local minima E Use interval notation to indicate where f x is concave down Concave down F List the values of all inflection points of f If there are no inflection points enter NONE values of inflection points G List the horizontal asymptotes of f If there are no horizontal asymptotes enter NONE

Consider the function f x 6x 6 List the values of the inflection points of f If there are no inflection points enter NOM

Consider the function f x x 4x 6 on the interval 0 4 Verify that this function satisfies the three hypotheses of Rolle s Theorem on the inverval f x is on 0 4 f x is on 0 4 and f 0 f 4 Then by Rolle s theorem there exists at least one value e such that f c 0 Find all such values c and enter them as a comma separated list Values of c

Biologists have observed that the chirping rate of crickets of a certain species is related to temperature and the relationship appears to be very nearly linear A cricket produces 120 chirps per minute at 70 F an 168 chirps per minute at 80 F a Find the linear equation that relates the temperature t and the number of chirps per minute n b If the crickets are chirping at 153 chirps per minute estimate the temperature in F Round your answer to the nearest degree X F

A spherical weather balloon is being inflated The radius of the balloon is increasing at the rate of 2 cm s Express the surface area of the balloon as a function of time t in seconds Let S 0 0 S t cm

Express the function G in the form f g There is more than one correct answer The function is of the form y f g x Use non identity functions for f x and g x G x f x g x 1 x 1

Find f g h 0 G f x x 2 g x x h x x 2

Express the function F in the form f g Enter your answers as a comma separated list Use non identity functions for f x and g x F x x 5 8 f x g x

Consider the following functions f x 3x 5 g x 2x 1 Find f g x Find the domain of f g x Enter your answer using interval notation Find g f x Find the domain of g f x Enter your answer using interval notation Find f f x Find the domain of f f x Enter your answer using interval notation Find g g x Find the domain of g g x Enter your answer using interval notation

Use f x 3x 5 and g x 2x to evaluate the expressio a f f x b g g x

Find f g f g fg and f g and their domains f x x x g x x Find f g x Find the domain of f g x Enter your answer using interval notation Find f g x Find the domain of f g x Enter your answer using interval notation Find fg x Find the domain of fg x Enter your answer using interval notation Find 1 x Find the domain of x Enter your answer using interval notation

Use f x 2x 3 and g x 5x to evaluate the expression a fog 2 b g f 2

Consider a triangle ABC like the one below Suppose that a 37 b 26 and c 36 The figure is not drawn to scale Solve the triangle Carry your intermediate computations to at least four decimal places and round your answers to the nearest tenth If there is more than one solution use the button labeled or A 0 B C 0 or X No solution S

By definition f g x f g x So if g 7 11 and f 11 20 then f g 7

From the graphs of f and g in the figure we find the following Assume that each point lies on the gridlines f g 2 f g 2 fg 2 1 2 10 5 bo g y 10 5 5 10 f 5 10 X

stone is dropped in a lake creating a circular ripple that travels outward at a speed of 65 cm s a Find a function g that models the radius as a function of time t in seconds g t b Find a function f that models the area of the circle as a function of the radius r in cm f r c Find f g f g What does this function represent This function represents the area of the ripple as a function of the radius This function represents time as a function of the area of the ripple This function represents the radius of the ripple as a function of time This function represents the area of the ripple as a function of time This function represents the radius of the ripple as a function of area

Consider a triangle 4 BC like the one below Suppose that 4 34 b 67 and c 3 The figure is not drawn to scale Solve the triangle Carry your intermediate computations to at least four decimal places and round your answers to the nearest tenth If there is more than one solution use the button labeled or 0 0 a B or X No solution S

4 Plot the following functions a f x x 2x x 1 b g x x sin x 2 tanx 0 x 4 5 x 5

4 2 points Consider the curve a helix parametrized by r t cost sint at where a 0 is a positive constant As a increases does the curvature increa decrease or stay the same Justify your answer by describing what happens to curve and interpreting the formula K d where s d where s s t is the reparametrizat of r such that r s 1 You need not compute the curvature directly

4 Bonus 2 Find the circle passing through the three points 0 4 2 2 and 2 2

Find the area of the given triangle Round your answer to the nearest tenth Do not round any intermediate computations 37 15 square units

1 Find the derivatives of the following functions A 5 pts f x x 4x 3x 1 B 5 pts g x sin x cos x 2 5 pts Given the function h z 5z 10z 1 5 compute h 2 3 5 pts Suppose that f 5 2 g 5 1 5 2 and g 5 3 Find h 5 where h x 3 f x 9

X 4 3 2 1 0 1 2 3 4 4 3 2 1 0 1 2 3 4 f x f x f x f x Derivative Function 4 3 2 1 0 1 2 3 4 Derivative Function 4 3 2 x 1 0 1 2 3 4 f x X 4 3 2 1 0 1 2 3 4 f x 4 3 2 1 0 1 2 3 4 f x f x f x Derivative Function X 4 3 2 1 0 1 2 3 4 Derivative Function X f x 3 2 1 0 1 2 3 4 f x

Evaluate the indicated limits by evaluating the function for values shown in the table and observing the values that are obtained Do not change the form of the function a Find lim x b f x I 1 x Find lim I 0 X ex X 1 1 0 900 0 1 0 990 0 01 0 999

Show all your work and clearly box the final an 1 Use the following graph of f x to find each limit 8 pts 5 Find lim f x x 2 c Find lim f x x 2 2 7 d Find f 2 1 2 3 4 a b C 2 Graph the following function to find the indicated limits and function values 13 pts a Find lim f x x 2 d lim f x lim f x S lim f x lim f x X 4 3 for x 2 x 1 for x 2 f x 2x 3 N 65432 1 7 6 5 4 3 2 1 1 123456 2 3 4 5 6 1234567 e Is the function continuous or discontinuous at x 2 Explain your answer using the definition of a continuous function Hint Verify the three conditions we discuss in class 4 pts

Soda Corporation purchased 2 800 shares of its own 0 50 par value common stock for 165 000 As a result of this transaction the company s stockholders equit OA increased by 163 600 B increased by 165 000 OC increased by 1 400 D decreased by 165 000

a Determine the equation of the line that passes through A 2 5 and is parallel to the line with an equation of 3x 2y 5 b Determine the equation of the line perpendicular to the line 3 having equation y x 6 and passing through the point 4