Calculus Questions

1 Solve the following equations round to 2 decimal places where appropriate a 3 12 b 10 2 4x 2

1 Compute the value of f x 7 3xy 5 dr if C is the curve consisting of the circle centered at the origin of radius 3 and the line y x as depicted in the picture 3

For the functions f x 6 x and g x x 4x 60 find f g f g fg and Determine the domain for each function g What is the domain of fg Select the correct choice below and if necessary fill in the answer box to complete your choice OA The domain of fg is B The domain of fg is Use a comma to separate answers as needed OC The domain of fg is O O A Type your answer in interval notation x Simplify your answer f What is the domain of Select the correct choice below and if necessary fill in the answer box to complete your choice g B f The domain of is g Use a comma to separate answers as needed f

1 Evaluate lim f x where x 0 f x sin 2x if x 1 n for some non zero integer n x 1 x otherwise Here x denotes the integer closest to x

For the following vector convert from x and y coordinates to magnitude and direction x 5 y 5 r 0

Given the vector u 67 3j find the magnitude and angle in which the vector points measured counterclockwise from the positive x axis 0 0 2

Find c if a 2 51 mi b 3 51 mi and ZC 40 2 degrees CROBRED mi Assume ZA is opposite side a LB is opposite side b and ZC is opposite side c

Given 8 7 and b 10 3 find 4c 6b 4c 6b

Find the angle where 0 0 180 between the following vectors Give your answer accurate to at least 2 decimal places u 117 6j 127 137 0

For the following vector convert from magnitude and direction to x and y coordinates r 3 0 20 X y

Let a 57 j and b 7i j Find a b

Write the product as a sum or difference 10 cos 22w sin 16w

If sin x in Quadrant 1 find 27 28 tan 2x Please enter answer accurate to 4 decimal places

The table below gives the number of calories for each type of candy in a certain box These are the only types of candy in the box Type of candy Number of calories Caramel 300 Peppermint 150 Cinnamon 250 Chocolate 250 200 Butterscotch A conditional statement is given below Give the converse contrapositive and inverse of the statement Also use the table above to decide if each statement is true or false Assume that any candy mentioned comes from the box Given statement Converse Contrapositive Inverse If a candy is chocolate then the candy has 250 calories True If Choose one then Choose one If Choose one then Choose one If Choose one then Choose one True O False O True False O True False O False X

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

The Jones Company has just completed the third year of a five year MACRS recovery period for a piece of equipment it originally purchased for 295 000 a What is the book value of the equipment The book value of the equipment after the third year is Round to the nearest dollar b If Jones sells the equipment today for 175 000 and its tax rate is 35 what is the after tax cash flow from selling it The total after tax proceeds from the sale are Round to the nearest dollar c Just before it is about to sell the equipment Jones receives a new order It can take the new order if it keeps the old equipment Is there a cost to taking the order and if so what is it Explain Assume the new order will consume the remainder of the machine s useful life A Yes the cost of taking the order is the lost 50 976 in book value B No Jones already owns the machine so there is no cost to using it for the order c Yes the cost of taking the order is the extra depreciation on the machine D Yes the cost of taking the order is the lost after tax cash flow of 131 592 from selling the machine

Determine a general formula or formulas for the solution to the following equation Then determine the specific solutions if any on the interval 0 2 sin 0 cos 0 2 Describe in the most concise way possible the general solution to the given equation where k is any integer Select the correct choice below and fill in all the answer boxes within your choice Simplify your answers Type any angle measures in radians Use the smallest non negative angle possible when describing the general form of each angle Use ascending order entering the expression based on the smallest angle first Type exact answers using as needed Use integers or fractions for any numbers in the expressions OA 0 OB 0 O c 0 O D 0 k k or 0 k or 0 k or 0 E There is no solution k k or 0 k k or 0 k or 0 K k

Tolo Co plans the following repurchases 9 6 million in one year nothing in two years and 20 6 million in three years After that it will stop repurchasing and will issue dividends totaling 24 4 million in four years The total paid in dividends is expected to increase by 2 9 per year thereafter If Tolo has 2 3 million shares outstanding and an equity cost of capital of 10 9 what is its price per share today The stock price would be Round to the nearest cent

6 Anle Corporation has a current stock price of 16 58 and is expected to pay a dividend of 1 20 in one year Its expected stock price right after paying that dividend is 18 81 a Anle s equity cost of capital is Round to two decimal places b The portion of Anle s equity cost of capital that is expected to be satisfied by the dividend yield is Round to two decimal places The portion of Anle s equity cost of capital that is expected to be satisfied by capital gains is

Daily Enterprises is purchasing a 9 8 million machine It will cost 49 000 to transport and install the machine The machine has a depreciable life of five years and will have no salvage value The machine will generate incremental revenues of 4 2 million per year along with incremental costs of 1 3 million per year If Daily s marginal tax rate is 35 what are the incremental earnings net income associated with the new machine The annual incremental earnings are Round to the nearest dollar

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

Solve for the exact solutions in the interval 0 2 List your answers separated by a comma if it has no real solutions enter DNE 3 sin 2 2 3 sin 3

Solve for the approximate solutions in the interval 0 27 List your answers separated by a comma round to two decimal places If it has no real solutions enter DNE 2 cos 0 2 cos 0 1 0

Directions Use your knowledge of geometric figures to set up an equation and solve for the missing variables X 400 ft v 36

Directions Draw a diagram create an equation and then solve for the desired measurement Kami is standing 12 feet from a rock climbing wall When she looks up to see her friend ascend the wall the angle of elevation is 56 How high up the wall is her friend V HINT Round to the nearest thousandths if necessary

4 How can the concept of families of polynomials help us analyze and compare different type of functions

1 Perform long division Express the answer in quotient form and write the statement that could be used to check the division x 9x 5x 3 divided by x 2

Substituting and combining like terms we get 4f x 2f x 2f x Part 3 of 3 Therefore f x aex 2 be x Submit Skip you cannot come back 0 ex 2 Select a solution to the differential equation 4y 2y 2y 0 0 ex

3 Find an equation for the following polynomial 34 16 3 2 1 0

We have f x Part 2 of 3 ae 2 be 1 Submit Skip you cannot come back ac be and f x ae Substituting and combining like terms we get 4f x 2f x 2f x 4 be ex 2 ae be X

Exercise b If 1 and 2 are the values of r that you found in part a show that every member of the family of functions y ae 1x be2x is also a solution Part 1 of 3 We must decide whether 4F x 2f x 2f x 0 for f x aex 2 be x We have f x and f x

2 ex 4r 2r 2 r V Part 3 of 3 From the previous step we have ex 42 2r 2 0 Since ex is never 0 then we know 4 2r2 0 Solving the above equation gives the solutions to the quadratic equation smaller value larger value L 11 47 2r 2

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

a What can you say about a solution of the equation y 1 3 y2 just by looking at the differential equation O The function y must be strictly decreasing on any interval on which it is defined The function y must be increasing or equal to 0 on any interval on which it is defined O The function y must be strictly increasing on any interval on which it is defined The function y must be equal to 0 on any interval on which it is defined The function y must be decreasing or equal to 0 on any interval on which it is defined b Verify that all members of the family y 3 x C are solutions of the equation in part a x c y X C LHS y Y x c X C d Find a solution of the initial value problem y 1 3 y y 0 0 1 1 1 1 R RHS c Can you think of a solution of the differential equation y 1 3 y that is not a member of the family in part b Oy x is a solution of y 1 3 y2 that is not a member of the family in part b B O Every solution of y 1 3 y is a member of the family in part b O y 0 is a solution of y 1 3 y2 that is not a member of the family in part b O y x is a solution of y 1 3 y that is not a member of the family in part b O y 3 is a solution of y 1 3 y2 that is not a member of the family in part b

Which of the following functions are solutions of the differential equation y y sin x Select all that apply O y cos x Oy x sinx 5x cos x O y sin x Oy x sin x Dy exco COS X

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

Find the average value of the function f x a 1 x on the interval 0 1 0 2 2 1 ONE 2 2 1 0 1 2 2

Find the arc length of the graph of the function f x over 0 1 0 2 2 1 OM 6 1

Find the length of the curve with parametric equations z 2t y 3t where t is from 0 to 1 O 2 2 1 04 2 1 08 2 4 04 2 2

Set up a definite integral for the length of the curve with parametric equations 2t y 3t where t is from 0 to 1 a You must show your work to receive full credit

Set up a definite integral for the volume of the solid generated by revolving the plane region bounded by y 1 y 0 x 0 x 1 about the y axis You must show your work to receive full credit

t up a definite integral for the volume of the solid generated by revolving the plane region bounded by y x y 0 x out the x axis You must show your work to receive full credit Edit Format Table

Set up a definite integral for the area of the region bounded by the graphs of y x y x in the first quadrant You must show your work to receive full credit Edit Format Table

Find the area of the region bounded by the graphs of y x y in the first quadrant O 6 4 2 3 0 4 2 6 3 0 1 O

Find the area of the region under the graph of y and above the interval 1 3 223 320 81

Consider the surface defined by the collection of points x y z R such that 3xz5 xy z 4 Show that z f x y near the point 1 0 1 After this compute the gradient vector V 1 0

O 2 4 5 3 2 0 5 1 23 5 NOTE AB is not possible In order to multiply matrices the number of columns from the first matrix must match the number of rows from the second matrix Question Suppose C Find CD 5 4 0 BA 2 5 51 3 9 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 6 5 6 4 1 1 c 2 3 and D 9 1 3 4 5 4

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

Magnitude of a Vector The magnitude of a vector v is the length of that vector Example 2i 3j 2 3 4 9 13 Question Find 7i 4j For more information on this topic look in your textbook at section 6 6 You might also want to look at the Vectors presentation 17 33

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