Differentiation Questions and Answers

An inverted conical water tank with a height of 16 ft and a radius of 8 ft is drained through a hole in the vertex at a rate of 5 ft s see figure What is the rate of change of the water depth when the water depth is 2 ft Hint Use similar triangles 8 ft 16 ft Let V be the volume of water in the tank and let h be the depth of the water Write an equation that relates V and h Type an exact answer using as needed

n 1 n 1 2 4 3 4 5 6 7 8 9 10 6 4 2 Sn MIRAS 6 2500 Graph both the sequence of terms and the sequence of partial sums on the same screen y y 4 68750 5 07813 4 98047 5 00488 4 99878 5 00031 4 99992 5 00002 5 00000 6 4 42

Part 1 of 5 9 10 To see 9 10 100 9 100 9 1000 81 1000 81 as a geometric series we must express it as 00 00 n 1 ar am n 0 For any two successive terms in the geometric series arn 1 the ratio of the the two terms arn arn 1 I simplifies into an algebraic expression given b

population is 5000 Find an expression for the number of bacteria after t hours OP 100 25t OP 100e OP 100e In 25 t OP 5000et Question 8 In 50 Continuing from the bacteria culture problem find the number of bacteria after 4 hours 250000 O 200000 O 100000 10 pts

A bacteria culture starts with 100 bacteria and the growth rate is proportional to the number of bacteria After 2 hours the population is 5000 Find an expression for the number of bacteria after t hours OP 100 25t In 50 OP 100e 2 OP 100e In 25 t O P 5000et t

All members of the family y Ce are solutions of the differential equation y 2xy True False

Question 1 The function y x sin x is a solution of the differential equation y y 2 cos x True 2x sin x

Consider the scalarfield z f x y 0 5x y xy xy defined on the triangle with the corners A 2 2 B 4 1 and C 2 1 Determine the global extrema of the scalarfield

a r t b r t and c r t r t 19 r t t i 3 Answer a 3t i tj b 6ti j c 18t t

entiation of Vector Valued Functions In Exercises 3 4 5 6 7 8 9 and 10 find r t r to and r to for the given value of to Then sketch the curve represented by the vector valued function and sketch the vectors r to and r to 3 r t 1 t i tj to 3 r Answer 4 r t 1 t i t j 5 r t cos ti sin tj Answer Answer 6 r t 3 sin ti 4 cos tj to 7 r t e e to 0 Answer r t 2 sin ti 2 cos tj k 3 2 8 r t e t et to 0 9 r t 2 cos ti 2 sin tj tk 3 2j to 1 2i k 3 2 to 7 2 2 k KN to 3 2

2 Bessel functions are useful in the analysis of problems that involve oscillations Very impor tant equations in differential equations involve Bessel functions in their solutions If a is a positive integer the Bessel function Ja r of the first kind of order a is defined by the power series Ja 3 n a 2 n 0 Show that this power series is convergent for every real number x 2n a

16 X Find an equation of the curve that passes through the point 0 4 and whose slope at x y is x y is X

c O 4 2 4 2 2 4 6 2 4 X 4 2 6 4 2 1 2

Find the cofactor of each element in the second row 2 6 3 15 1 1 1 5 The cofactor of 1 is

Higher Order Differentiation In Exercises 19 20 21 and 22 find a r t b r t and c r t r t 19 r t t i t j 2 Answer 20 r t t i t t j 1

Finding a Derivative In Exercises 11 12 13 14 15 16 17 and 18 find r t 11 r t t i 5tj Answer 12 r t ti 1 j 13 r t 3 cos ti 2 sin tj k

Integrating both sides gives us In 7 75 Part 3 of 5 We now know that T 75 ekt C 75 Kekt Since T 0 185 then K In T 75 kt C which becomes T 75 kt C 75 kt C

Therefore an antiderivative of f x x7 10 is found as follows x17 10 F x x17 10

Evaluate the definite integral 1 2 6 21 dr 1 2

A graphing calculator is recommended Find the general indefinite integral Use C for the constant of integration cos x 2x dx Illustrate by graphing several members of the family on the same screen 10 5 y 20 15 10 C 10 C 5 C 0 10 5 10 y 15 10 10 5 f 1 10 C 11

Evaluate the integral R 6 S sec 0 tan 0 de

An antiderivative of x as long as n 1 is

Alternative Form of Dot Product In Exercises 19 and 20 use the alternative form of the dot product to find u v 19 u 8 v 5 and the angle between u and v is 3

y t 50e t 30 In 2 50 2 50 Exercise b How much of the sample remains after 40 years 50 2 Step 1 After 40 years we have the following y 40 50 2 mg mg Round your answer to two decimal places

When solving the following equation using adjusted interval methods cos 0 over the interval 0 4 what is the ADJUSTED INTERVAL that you would use to solve 0 8T None of these 0 2x 0 4 0

Find the area of the region that is bounded by the given curve and lies in the specified sector st 2 r 180 0 es

Find the Maclaurin series for f x using the definition of a Maclaurin series Assume that f has a power series expansion Do not show that R x 0 f x In 1 5x f x 00 R n 1 Find the associated radius of convergence R 1 n 1 x 1 5 1 n 15 11 10 012 X

2 Ignoring the constant of integration integrating the left side gives zu 2u du u Part 3 of 5 1 Ignoring the constant of integration integrating the right side gives 2t sec t dt

Question 9 R duis l expression ci dessous l aide de la notation exponentielle sans d exposants n gatifs 27 5 C

A rickshaw company counted 39 ticket receipts last week The price for a weekday ticket is 7 and the price for a weekend ticket is 9 50 The rickshaw driver collected a total of 333 for the week Let x represent the number of weekend tickets and y represent the number of weekday tickets Which system of equations represents the situation Hint When coefficients contain decimals that can be easily converted to whole numbers it s good form to convert them This can be done by multiplying both sides of the equation by the same number that will clear the decimal Select the correct answer below y x 39 O O 14y y 15y y 15y y 14y y 7y 19x 666 x 39 19x 666 x 39 19x 666 x 39 19x 666 x 39 9x 333

A company sells widgets The company s fixed and variable costs are modeled by the function C x 0 74x 25000 Their revenue is modeled by the function R x 2 3x How many widgets do they have to sell to break even Round your answer up to the nearest whole number and do not include units Provide you olour

A concert manager counted 33 ticket receipts the day after a concert The price for an adult ticket was 14 and the price fo a student ticket was 9 50 The register confirms that 403 50 was taken in How many adult tickets and student tickets were sold Provide your answer below

The range of h x X 2x 1 1 3 x 21 O A 0 1 B 0 3 C 3 00 D All real numbers D E 0 3 F 0 1 is

Find the solution to the following system of equations by graphing 3x y x y Drag the movable points to graph each of the linear equations The plot automatically calculates and shows the coordinate of the intersection of the two lines Provide your answer below 10 5 10 5 0 2 0 3 3 5 10

The adjacent table gives the number of worldwide Internet users in millions Complete parts a through b Year 1995 1996 1997 1998 1999 2000 2001 Users millions 11 39 77 189 292 341 521 Year 2002 2003 2004 2005 2006 2007 Full data se b Use the model to predict the number of users in 2011 million The number of users in 2011 will be approximately Round to the nearest whole number as needed Users millions 589 705 821 1010 1099 1235 a Find the cubic function that is the best fit for the data with y equal to the number of millions of users and x equ the number of years from 1990 y Use integers or decimals for any numbers in the expression Round to three decimal places as needed

Using Lagrange multipliers find the min and max distance from the point P 0 1 0 to the ellipsoid 2x y z 6 and also find where the min max distance is attained on the ellipsoid You can assume that the absolute extrema must exist and be attained only at the candidate points determined by Lagrange multipliers

2 f x 4 x OA Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The relative minimum point s is are and the relative maximum point s is are Simplify your answer Use integers or fractions for any numbers in the expression Type an ordered pair Use a comma to separate answers as needed and there are no relative maximum points B The relative minimum point s is are Simplify your answer Use integers or fractions for any numbers in the expression Type an ordered pair Use a comma to separate answers as needed OC The relative maximum point s is are and there are no relative minimum points Simplify your answer Use integers or fractions for any numbers in the expression Type an ordered pair Use a comma to separate answers as needed OD There are no relative minimum points and there are no relative maximum points Choose the correct graph below X Q the function Q O B X CIE 10 Q O C Q Q O D 10

4 How do exponential and logarithmic functions relate to each other

The following graph contains the pure form of an 8 Hz cycles per second wave for a time interval of one second What is the period 0 5 0 5 8 O 0 125 2 I 0 2 0 4 X 0 6 0 8

The angle of eleveation and depression are always measured from a line prallel to the ground O True False

in x 2 sin x for all values of x True False

Which of the following pairs express the same graph in 20 27 8 x Of x 3sin 20 x g x 3cos 20 f x 3sin 20 g x 3cos 20 f x 3sin 0 8 x 3eus 3cos Of x 3sin 0 x g x 3cos e 2

In a recent international sport competition the top three countries Country A Country B and Country C won a total of 127 medals Country B won 11 more medals than Country C Country won 41 more medals than the total amount won by the other two How many medals did each of the top three countries win

8 Gavin has purchased a car He financed a loan of 25 697 at 8 9 compounded monthly He considers repaying the loan in 36 or 48 monthly payments His first payment is 1 month from his date of purchase 6 marks a Determine the monthly payment for a 36 month loan b C Determine the monthly payment for a 48 month loan Determine the interest Gavin saves if he pays off the loan in 36 months

Substituting for f x and its derivatives we have the following Simplify and factor out the exponential 0 4 ex 2rex 2e x ex

Part 1 of 3 To determine the values of r for which ex satisfies the differential equation we substitute f x e in the equat 4f x 2f x 2f x 0 Thus we need to find f x and f x f x ex f x F x

O D A matrix is a rectangular array of numbers The dimensions of a matrix are written as rows x columns 3 2 Example A 2 0 and B 1 5 3x2 matrices Question Use the two matrices from above to find 2A B 3 4 0 3 12 For more information on this topic look in your textbook at section 8 3 You might also want to look at the Matrix Operations presentation 2 4 3 2 4 3 2 3 0 8 4 7 0 0 1 2 are both 0 12

Recall the equation for a circle with center h k and radius r At what point in the first quadrant does the line with equation y 0 5x 1 intersect the circle with radius 3 and center 0 1 y

Find if possible a AB and b BA if A AB 15 5 17 3 5 5 3 2 4 2 15 4 5 3 3 2 3 a Select the correct choice below and if necessary fill in the answer boxes to complete your choice 7 Simplify your answers and B OA BA Simplify your answers B This mix operation is not possible 1 5 21 01 5 33 B This matrix operation is not possible b Select the correct choice below and if necessary fill in the answer boxes to complete your choice

Triangle ABC shown below has mZA 92 b 10 and c 14 Find the Round your answer to the nearest tenth and do not include units in your an A 10 92 14 a