Calculus Questions

9 points Let S t be the function which specifies the distance in km from a runner to the start line at time t in hours of a race The graph of S t is given below 4 3 2 1 y km 2 3 b What happened between times 3 and 4 S t Use the graph to give approximate answers to the following problems a When was the runner running the fastest c What is the meaning of S 1 t h

11 Approximate da using a Riemann sum with n 3 terms and the midpoint rule What does this number have to do with In 4 12 The velocity of a particle is given by v t 1 1 m s a Find the acceleration a 2 of the particle W How for does the particle travel in the first 5 seconds

9 points Find the dimensions of a can i e a cylinder of radius r height and volume 1 m with minimal surface area Hint the volume V nr h an the surface area S 2 rh 2 r

A ball is thrown into the air by a baby alien on a planet in the system of Alpha Centauri with a velocity of 46 ft s Its height in feet after t seconds is given by y 46t 17t A Find the average velocity for the time period beginning when t 1 and lasting for the following number of seconds 01 sec y 10 1 y 1 0 01 X syntax incomplete Hint If the throw starts at t 1 and lasts 01 sec what will the ending time t2 be 005 sec 45 1 X 002 sec 46 4 001 sec 45 83 X NOTE For the above answers you may have to enter 6 or 7 significant digits if you are using a calculator Estimate the instanteneous velocity when t 1 12 X

Consider the function y f x graphed below Give the

Suppose the rate of growth of bacteria in a Petri dish is given by q t 3 where t is given in hours and q t is given in thousands of bacteria per hour If a culture starts with 8000 bacteria find a function Q t that gives the number of bacteria in the Petri dish at any time t How many bacteria are in the dish after 2 hours You should round your answer to the nearest whole number Do not include any commas in your final answer if applicable Provide your answer below

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Prof Rudy Meangru Solve the following equ 1 sin 1 2

8 sec 0 tan 0 3

Compute the derivatives of the functions arctanx a y 2arctana In2 2 1 x 1 x 4

2 BUSINESS REVENUE If a company s marginal revenue function is MR z xe 4 find the revenue functic Hint Evaluate the constant C so that revenue is 0 at x 0 X 4 xe dv U EX du 1 dx 4 4 x du e v e

Which of the following functions satisfy the given differential equation y y a ex b xe x c 3e x d ex e xex

1 Use integration by parts to find the integral In x ln x dx

E GENERAL PERMANENT ENDOWMENTS Find the size of the permanent endowment needed to generate an T yeart at continues rater annual 12 000 forever at a continuous interest rate of 6 Present value TC t e rt dt C Continous stream years

se integration by parts to find the integral x ln x dx C

press 1 234234234234 as a ratio of intege

3 Express 7 45 as a ratio of integers

Find the volume V of the described solid S The base of S is the region enclosed by the parabola y 2 3x2 and the x axis Cross sections perpendicular to the y axis are squares V

Given the graph below solve the following inequality h x 20 10 1 x 9 8 7 30 9 8 7 6 5 3 2 1 2 6 8 9 10 6 7 8 9 10

Que Find the radian measure that corresponds to the given degree measure 225 a 40500m Obl 54 Od 5

Find the exact value of COS O 15 15 4 os sin 2 15 15 O 4 15

Find the value of COS given that sin a 4 6 11 b 4 6 5 6 TT and ta

Solve for 0 0 0 2x 2 sin 0 sin 6 31 21 5x 0 2 5 10

Solve for 2 372 1 81 Olog 82 7 28 7 5

Simplify the following trigonometric expression sect cos O al 1 b cote Od tan 0 d sin 0

oblem 5 For each change equation 1 find the equilibrium solution s if any exist 2 graph the equilibrium solution s along with the graphs of the graphs of y z for the given initial values dy y 3 y 0 1 and y 0 4 da dy b y 2 y 4 y 0 1 y 0 3 and y 0 5 da a

Identity the expression that completes the following equation so that it is sin z 1 cos z a 1 sin z CoS z Ob 1 cos x Od 1 sin r c cos z d sin a

Given side a 24 side b 28 and side c 30 find the area of triangle AABC 0316 336 909 285

Solve the following equation for x log11 x 5 4 8 O a x 126 b x 116 Oc x 2043 d x 24 2

Find the degree measure that corresponds to the given radian measure 5F O a 108 Ob 300 R Od 300 Q108

45t 6 A particle has a velocity modeled by v t 5t3 2t 4 What is the total distance traveled from t 1 tot 2 1 Provide your answer below measured in cm s over the interval 1 st 2 You may either enter an exact answer or round your answer to the nearest tenth of a centimeter

The price of a home is 270 000 The bank requires a 5 down payment at the time of closing The mortgage on the house is a 30 year fixed rate mortgage at 7 Use this information to find the amount of the monthly payment Round your answer to the nearest cent Use the formula below to solve the problem Payment 2 nxt 1 1 1 1 695 2 1 796 3 1 807 PL n 4 1 707

TABLE OF INDEFINITE INTEGRALS f cf x dx c f f x dx fkdx kx C Xn 1 fxndx C n 1 n 1 fex dx ex C sin x dx cos x C sec x dx tan x C sec x tan x dx sec x tan x dx sec x C S 1 x 1 dx tan x C sinh x dx cosh x C 18 f x g x dx f f x dx f g x dx dx In x C faxdx ax In a cos x dx sin x C fcsc x dx cot x C csc csc x cot x dx C x cot x dx csc x C CSC dx sin x C 1 x cosh x dx sinh x C sin 2x dx

TABLE OF INDEFINITE INTEGRALS f cf x dx c f f x dx fkdx kx C Xn 1 fxndx C n 1 n 1 fex dx ex C sin x dx cos x C sec x dx tan x C sec x tan x dx sec x tan x dx sec x C S 1 x 1 dx tan x C sinh x dx cosh x C 18 f x g x dx f f x dx f g x dx dx In x C faxdx ax In a cos x dx sin x C fcsc x dx cot x C csc csc x cot x dx C x cot x dx csc x C CSC dx sin x C 1 x cosh x dx sinh x C sin 2x dx

TABLE OF INDEFINITE INTEGRALS fcf x dx cff x dx fkdx kx C X 1 f xndx C n 1 dx In x C n 1 fe dx e C sin x dx cos x C sec x dx tan x C sec x tan x dx sec x tan x dx sec x C 1 x 1 sinh x dx cosh x C f x g x dx f f x dx g x dx dx tan x C ax a dx in a cos x dx sin x C fcsc x dx cot x C CSC x cot x dx csc C CSC X C 1 1 x S cosh x dx sinh x C enc dx sin x C 14 csc t 2e dt tai

Question A particle has a velocity modeled by v t 3 2ln t 5 measured in cm s over the interval 4 t 7 What is the total distance traveled from t 4 to t 7 You may either enter an exact answer or round your answer to the nearest tenth of a centimeter Provide your answer below

TABLE OF INDEFINITE INTEGRALS fcf x dx cff x dx fkdx dx kx C fxndx C n 1 dx dx In x C Xn 1 n 1 fe dx e C sin x dx cos x C sec x dx tan x C sec x tan x dx sec x C 1 x 1 sinh x dx cosh x C f f x g x dx f f x dx f g x dx dx tan x C ax a dx In a I cos x dx cos x dx sin x C C fcsc x dx cot x C S CSC x cot x dx csc x C x dx f cosh x dx dx sin x C sinh x C 10 v v 2 dv 2

TABLE OF INDEFINITE INTEGRALS f cf x dx cf f x dx kdx kx C Xn 1 fxndx C n 1 n 1 fe dx e C S sec x dx tan x C S sec x tan x dx sec x C 1 S x 1 sinh x dx cosh x C sin x dx cos x C dx tan x C f x g x dx f f x dx f g x dx d dx n x C ax fa dx C In a f cos x dx cos x dx sin x C fcsc x dx cot x C CSC x cot x dx csc x C csc x cot x dx 1 S v x f cosh x dx sinh x C dx sin x C 6 x 3 x dx W x3 2

A worker has developed a process for assembling a new product Initially assembly will be slow because the process is new but eventually the process will allow for close to 90 items to be produced a day This can be modeled by the equation f x 90 110 x 3 where f x represents the rate of production in items per day How many items will be produced from day one to day two Round your answer to the nearest whole item Provide your answer below

Evaluate the limit lim 3 tan 4x Enter a simplified multiple of or or 3n 2 X

Given the absolute function graphed below find the following 4 a Domain b Range Increasing 6 S Decreasing 4 3 O 7 4 2 a 4 2 c On what interval is the function increasing and on what interval is the function decreasing 4

Evaluate the limit lim 3 tan 4x Enter a simplified multiple of or or 81X 3x 2 X

Consider a 400 N weight suspended by two wires as shown in the figure Find the magnitudes and components of the force vectors F and F2 F N Round to one decimal place as needed F N Round to one decimal place as needed F Round to one decimal place as needed F F14 25 50 400 N

Find the volume of a parallelepiped with one of its eight vertices at A 0 0 0 and three adjacent vertices at B 2 1 0 C 0 5 1 and D 4 6 8 The volume of the parallelepiped is Simplify your answer

Suppose that a box is being towed up an inclined plane as shown in the figure Find the force w needed to make the component of the force parallel to the inclined plane equal to 2 2 lb The force w is about lb Round to the nearest hundredth as needed COO

Find the area of the parallelogram whose vertices are given below A 2 3 B 2 0 C 8 3 D 4 6 The area of parallelogram ABCD is square units

Consider a 400 N weight suspended by two wires as shown in the figure Find the magnitudes and components of the force vectors F and F2 F N Round to one decimal place as needed F ON Round to one decimal place as needed F 00 Round to one decimal place as needed F1 25 50 400 N

A line passes through the points P 1 7 8 and Q 10 2 2 Find the standard parametric equations for the line written using the base point P 1 7 8 and the components of the vector PQ x 0 y 0 z 0 X Type expressions using t as the variable COLO

Find the following for the vectors v 101 9k and u i k a v u v ul b the cosine of the angle between v and u c the scalar component of u in the direction of v d the vector proj u a v u Type an exact answer using radicals as needed v Type an exact answer using radicals as needed u Type an exact answer using radicals as needed b The cosine of the angle between v and u is Type an exact answer using radicals as needed c The scalar component of u in the direction of v is Type an exact answer using radicals as needed d proj u Type your answer in terms of i j and k Type an exact answer using radicals as ne

Write inequalities to describe the following sets a The interior of the sphere of radius 4 centered at the point 3 6 2 b The exterior of the sphere of radius 4 centered at the point 3 6 2 a Choose the correct inequality OA x 3 y 6 z 2 s1 OB x 3 y 6 z 2 16 OC x 3 y 6 z 2 16 OD x 3 y 6 z 2 10 16 b Choose the correct inequality OA x 3 y 6 z 2 16 OB x 3 y 6 2 2 16 OC x 3 y 6 z 2 16 16 OD x 3 y 6 z 2 1 2