Calculus Questions

Consider the equation 2x 11x 7x 6 A student performed synthetic division on this polynomial and their work is shown below 6 11 7 12 6 6 2 1 1 0 Based on their work which is correct write the equation of the polynomial in factored form 6 2

Consider the polynomial x 4x 7x x 12 Using Decartes Rule of Signs create a table like the one below and fill in the possible combination of roots Positive 3 Negative Imaginary

Given the polynomial x3 7x2 16x 12 answer each of the follwing Be sure to label your answer by labeling the parts a b c etc To get full points show your work for this problem on your scratch paper a State the y intercept b State the degree and what it tells you about the graph of the function c Using Decartes rule of signs state the number of possible real roots zeros d Use the rational root theorem to state all possible rational roots e Using either long division or synthetic division break down the polynomial into factored form Clearly state your result

Us eproperties of logatithms to expand the logarithmic expression as much as possible m n k 1085 1 b 3 7 a 1 Oa log m log n 210g k Oblog m log n log 4 Oc log m log m 2log 4 3 O d 3log m 7log n 2log

State the different methods one can use to solve a system of equations Then explain which method you prefer and

Write the form of the partial fraction decomposition of the rational expression It is not necessary to solve for the constants 4x 3 x 9x 20 O a A x 4 B x 5 C x 9x 20 O b A x 4 B x 5 Cx D x29x 20 OC A x 4 B x 5 C x 4 x 5 O d A x 4 B x 5

Consider the function x2 8x 7 Which of the following is correct O vertex 4 9 intercepts 7 0 1 0 0 7 O vertex 4 9 intercepts 7 0 1 0 0 7 O vertex 4 9 intercepts 7 0 1 0 0 7 O vertex 4 9 intercepts 7 0 1 0 0 7

Match the graph with the appropriate function H 2 2 2 y f x x 2x 1 Of x x 2x 1 Of x x 2x 2 Of x x 2x 2 2 X

If a polynomial has a degree of 6 then the number of turns could be 3 O True False

14 Find the area of the shaded region on the right 16 0 y 4x y 48 16x 3 X

Find equations of all lines having slope 3 that are tangent to the curve y 3 x 9 Select the correct choice below and fill in the answer box es within your choice OA There is only one line tangent to the curve with a slope of 3 and its equation Type an equation OB There are two lines tangent to the curve with a slope of 3 The equation of the line with the larger y intercept is Type equations

Graph of the function f x log x You can only graph integer points for example you cannot plot the point 1 There is one point that can be moved above and below the x axis as needed Do not move either point off of the graph as you will not be able to click on the point again Provide your answer below 3 0 2 3 A 5 6

Using Descartes Rule of Signs write the greatest possible number of negative roots for the function g x 5x 7x 2x 2 holow

xpress the function as the sum of a power series by first using partial fractions Give your power series representation centered at x 0 2x6 6x 5 x DO f x C n 0

A ball is thrown eastward into the air from the origin The initial velocity is 45i 75k measured in feet per second The spin on the ball gives an acceleration of 5 ft s 2 northward Where does the ball land and with what speed

1 point Consider the system of differential equations Rewrite as Z a11 a12 921 922 Then det 911 921 x 912 922 2 and the eigenvalues are A A general solution is ciet 2 I 2 2 c e 21 x 2 x 1 x x 2 x 5 x2 1X

Use Descartes Rule to determine which of the following is a possible number of solutions for the function f x x 2x x 1 Select all that apply 3 positive 0 negative and 0 imaginary 1 positive 2 negative and 0 imaginary 2 positive 1 negative and 0 imaginary 1 positive 0 negative and 2 imaginary 0 positive 3 negative and 0 imaginary 0 positive 1 negative and 2 imaginary

Given the triangle below find cot 0 Enter your answer as a ratio of sides 24 25 7

The matrix A 26 21 28 23 Find its eigenvectors For 5 the eigenvector is and for 2 the eigenvector is has eigenvalues 5 and 2

1 point Given the matrix 3 3 2 2 2 3 1 30 find its determinant Do not use a calculator The determinant is

1 point Find the determinant of the matrix det A 3 A 341 9 2

Given the triangle below find sec 0 Enter your answer as a ratio of sides 0 41 40 9

Find the angle in degrees between 0 and 360 which intersects the unit circle at the point 1 Do not include the degree symbol in your answer For example if the answer was 20 you would enter 20 power below

Using Descarte s Rule of Signs which are possible combinations of zeros for f x 2x4 8x 12x 16x 10 Select all that apply 4 positive 0 negative 0 complex 2 positive 0 negative 2 complex 0 positive 4 negative 0 complex 2 positive 2 negative 0 complex 0 positive 0 negative 4 complex

Graph the function f x log x There is one point that can be moved You can only graph integer points for example you cannot plot the point above and below the x axis as needed Do not move either point off of the graph as you will not be able to click on the point again Provide your answer below 2 1 2 0 2 2 3 4 5 6

1 Find all solution in 0 2x a cos2x sinx 1 b cos x 2 cos x 3 c cos2xsinx sinx 0 d sec x 2tan x 0

2 Proof the following cos2x 1 a tanx sin2x 1 cos2x 1 cos2x tan x cos2x 1 sin2x cos2x 1 sin2x d 4 cos x cos2x 5 6sin x b

Here is a graph of the function f 5 3 2 2 3 2 3 Use the graph to find the following If there is more than one answer separate them with commas a All values at which has a local minimum b All local minimum values of f

Draw the function f x by dragging the blue points to the locations on the x axis that correspond to the zeros of the function Use the red point as a slider to select the number of zeros that your graph will show You may click on a point te verify its coordinates Provide your answer below 10 O zeros 4 5 40 8 30 20 10 0 10 20 A f x 4 x 2 x 1 x 3 5 10

1 Given that f x x 2x and that g x x 15

Find the equation of the exponential function whose graph is shown below Provide your answer below 12 18

Find the possible numbers of positive and negative real zeros for the function f x 2x4 10x 11x 15x 12 1 Select the correct answer below 4 2 or 0 positive real zeros 4 2 or 0 negative real zeros 4 2 or 0 positive real zeros 0 negative real zeros 0 positive real zeros 4 2 or 0 negative real zeros 0 real zeros

The function y e 5x is vertically stretched by a factor of 6 reflected across the y axis and then shifted up 1 units find the resulting function Provide your answer below

Consider the function f x 2x4 2x 6x 7x 2 Using Descartes Rule of Signs what is the maximum possible number of positive roots

The function y e 3x is vertically stretched by a factor of 4 reflected across the y axis and then shifted up 5 units find the resulting function Sorry that s incorrect Try again

the f x by dragging the blue points to the locations on the x axis that correspond to the zeros of the function Use the red point as a slider to select the number of zeros that your graph will show You may click on a point to verify its coordinates Provide your answer below 10 5 200 100 0 100 f x 2 x 5 x 1 x 8 10

f t dt 2 and f t dt then 0 5 01 0 1 05 20 f t dt 3

Determine the convergence or divergence of the series 2 1 m 13m2 Explain why to receive full credit Edit Format Table

Find the interval of convergence for the power series 1 Q 1 1 1 1 1 1 1 1

Determine if the geometric series converges or diverges If it converges find the sum of the series 04 O converges sum is 4 O converges sum is 2 O converges sum is 6 Odiverges

Find the limit of the sequence an Ox 20

A bacteria culture starts with 1000 bacteria and the growth rate is proportional to the number of bacteria After 1 hour the population is 3000 a Find an expression for the number of bacteria after t hours b Find the number of bacteria after 2 hours O a P t 1000 4 b 16000 O a P t 1000 3 b 9000

True or false y x cosa is a solution of the differential equation y y 2 sin True False

Solve the differential equation 2yy xe Please show your work to receive full credit

Find the length of the curve with parametric equations 2t y 3t where 0 t 1 022 1

Find the area of the region bounded by the graphs of x y x y O 1 0 1 1 01

Determine whether the improper integral converges or diverges If it converges find its value fi dz O Diverges O Converge 1 O Converge 2 O Converge

Find the indefinite integral f O In C O In C O In C Oln C 3 x x 2 da

Find the indefinite integral f sin x cos adx O cos zsina C cos a sin a C O 0 240 O C

Evaluate the integral fx sin x da Cos 1 1 2 COS 2 O sin 1 O 1 cos 1 2