Calculus Questions

Find a particular solution to the following differential equation Do NOT solve the coefficients y 4 4y 3 7y 6y 2y et 4t sint

nd the mass and center of mass of the lamina that occupies the region D and has the given density function D is the triangular region enclosed by the lines x 0 y x and 2x y 6 p x y 7x m x y

3 Find equations of all lines having slope 3 that are tangent to the curve y x 5 Select the correct choice below and fill in the answer box es within your choice OA There are two lines tangent to the curve with a slope of 3 The equation of the line with larger y intercept is and the equation of the line with the smaller y intercept is Type equations

Find the following for the vectors v 6i 5k and u i j 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 1 1 Type an exact answer using radicals as needed u 3 Type an exact answer using radicals as needed 1 183 b The cosine of the angle between v and u is Type an exact answer using radicals as needed 1 61 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 needed

Find the distance from the point 0 0 7 to the line x 3t y 2t and Z 2t The cross product used when calculating the distance is 0 0 7 x 3 2 2 14 21 0 The magnitude of the cross product is 637 Type an exact answer using radicals as needed 0 Type an exact answer using radicals as needed The distance is

Find the points on the given curve where the tangent line is horizontal or vertical Write answer in correct order using order pairs to get the credit r 1 cos 0

Graph the following function Drag the movable red point to set the asymptotes for one period of the given function Also place the blue point at the correct set of coordinates You may click on a point to verify its coordinates Provide your answer below TT TT 2 7 6 5 4 3 2 14 0 1 2 3 4 TT 2 1 3TT 2 y 4 cot 2TT 5TT 2 3TT

Evaluate the integral 1 3 1 2 cos 2 cos t i 6 sin t j sec t k dt 1 2 cos t i 6 sin t j sec t k dt i j k Type exact answers

a Evaluate b For what values of a does converge L 1 1 I fe e dr 1 dr

Use Taylor series to evaluate the lim 1 In 1 8x cos 8x x lim X 0 O A 32 O B 68 O c 4 O D 40

Find the curve s unit tangent vector Also find the length of the indicated portion of the curve TU r t 3 cos t j 3 sin t k Osts Osts Find the curve s unit tangent vector T t j k Type exact answers using radicals as needed The length of the curve is Type an exact answer using as needed

Find equations of all lines having slope 2 that are tangent to 18 the curve y X 4

cylindrical coordinate integral 0 The value is Type an exact answer using as needed r dz dr de

X The equation r t 3 cos t i 3 sin t j 9t k is the position of a particle in space at time t Find the particle s velocity and acceleration vectors Then write the particle s velocity at t product of its speed and direction 2 The velocity vector is v t The acceleration vector is a t The velocity vector at t written as a product of the speed and direction is 1 00 Type exact answers using radicals as needed as a

On the axes below draw the graph of the function f x x 3 Provide your answer below 5 10 5 0 5 4 10

y 4 cot x 1 3 Drag the movable black point to set the left vertical asymptote and shift the function the red point to set the right vertical asymptote thereby setting the period of the function and the blue point at the correct set of coordinates You may click on a point to verify its coordinates Note that the two asymptotes can be moved independently of each other and that only one period of the function is shown Provide your answer below 2TT 3TT 2 TT 4 TT 2 3 2 0 0 0 1 2 4 1 1 2 02 TT 2 3TT 2 I

Given that the polynomial f x has 9 x intercepts which of the following most accurately describes the degree of f x Select the correct answer below The degree of f x is at most 8 The degree of f x is at least 10 The degree of f x is at most 9 O The degree of f x is at least 8 The degree of f x is at least 9 The degree of f x is at most 10

Given that the polynomial f x has 5 turning points which of the following most accurately describes the degree of f Select the correct answer below The degree of f x is at most 6 The degree of f x is at most 5 The degree of f x is at least 6 The degree of f x is at most 4 The degree of f x is at least 4 The degree of f x is at least 5

Given that csc x 2 6 what is csc x

Given that csc x 2 what is csc x

Set the point Provide your answer below 10 1 3 21 3 13 10 0 TT 11 3 51 3

Given that tan 0 4 and 0 is in Quadrant III find sec 0 and sin 0 Provide your answer below

For the function I X CSC X set the moving point to a value of y a such that it is not a possible value of f x Provide your answer below 2 T 1 1 1 TT 5 2 95 0 1 F L T F 2TT

2 Drag the black dot to shift your graph in the desired direction Use the blue draggable dot to change the period Drag the orange dot to change the amplitude and or reflect with respect to the x axis The horizontal distance between the vertical dotted green lines corresponds to one period Provide your answer below 2TT 11 31 2 2 1 0 0 2T 0 23T 3 41

If cos 0 0 and tan 0 0 in which quadrant does 0 lie Select the correct answer below O Quadrant I O Quadrant II O Quadrant III O Quadrant IV

Given that csc x 7 what is csc x

Analyze and sketch a graph of the function Identify any relative extrema points of inflection and asymptotes Use a graphing utility to verify your results If an answer does not exist enter DNE f x arccos maximum minimum point of inflection horizontal asymptote vertical asymptote x y x y x y 0000

The system of differential equations has a general solution where Re eta x t Re etb Re etc Im eta x t Im e b Im etc x 3 9 9 0 1 4 1 2 x 1 x t c x t C x 1 C3x3 t

The matrix A det 9 9 3 1 2 a11 a12 921 a31 0 1 2 4 V a13 922 a23 932 a33 1 2 1 2 9 a 1 2 b 2 c 0 9 a 4 b 1 2 c 0 which is the system a a b b where b The eigenvalue with positive imaginary part is For eigenvalue A the eigenvector has components a b and c that satisfy 3 a 0 b 1 c 0 i a b 301 9 12 9 4 1 2 has characteristic equation b i b 0 Choosing the first component of the eigenvector as a 1 this eigenvector is a 13 18 c 0 c 0 i c 0 i and c 27 i

The matrix A det 3 A 4 2 3 2 a11 a12 921 922 3 2 3 2 4 2 a 2x a 2 4 a 32 b 0 which is the system c 2 c 4 c 32 d 0 which is the system The eigenvalues are For eigenvalue 2 the eigenvector has components a and b that satisfy 3 a 2 b 0 For the eigenvector is G a 4 3 N b 0 b 0 For eigenvalue 22 the eigenvector has components c and d that satisfy 32 c 2 d 0 0 d 0 d 0 has characteristic equation

Question Given that cos 0 necessary Brovide swer below and is in Quadrant III what is sin 0 Give your answer as an exact fraction with a

12 Find the particular antiderivative given that f x 15 24 8 and f 1 7 13 The marginal cost of manufacturing x units of a product is given by

Question Provide your answer below 0 in x The figure above shows a right triangle with dimensions in inches If 0 27 what is the length of x to the nearest inch 43 in

Given that cos 0 and 0 is in Quadrant IV what is sin 0 4 17 17 Simplify your answer including rationalizing the denominator Provide your answer below

The graph of the function f x log7 x is stretched vertically by a factor of 6 reflected over the x axis reflected over th y axis and shifted down by 8 units Find the equation of the function g x described above Note If you are using log you need to type it in and use the subscript button on the keyboard There is no log button Provide your answer below g x

Use trigonometric ratios to determine the length of x in the right triangle below X 71 5 cm Round your answer to the nearest tenth and do not include x or the units in your answer Just enter the numeric value

d y Find an equation for the line tangent to the curve at the point defined by the given value of t Also find the value of dx T x 8 sint y 8 cost t 4 represents the line tangent to the curve at t 4 The equation Type an exact answer using radicals as needed The value of d y 2 T at t is 4 at this F

1 point Solve the heat problem THINK U Uxx 0 x u 0 1 0 u T 1 0 u x 0 cos x

Find the mass and center of mass of the solid E with the given density function p E is bounded by the parabolic cylinder z 1 y and the planes x 4z 4 x 0 and z 0 p x y z m x y z X

Given the triangle below find csc 0 Enter an exact answer in fraction form 12 13 5 0

What trigonometric function would you use based upon the given information in the triangle shown below Select the correct answer below O O sin B sin B cos B b a 100 cos B tan B 2 B a

Provide your answer below 6 Reflection 5 4 3 2 1 0 electing the correct reflection then move the vertical asymptote and the key point f x log4 x 3 1 2 3 4 5

The distance from the Sun to Earth is approximately 149 600 000 km The distance form the Sun to Mercury is approximately 57 900 000 km The elongation angle a is the angle formed between the line of sight from Earth to the Sun and the line of sight from Earth to Mercury Suppose that the elongation angle for Mercury is 20 Use this information to find the two possible distances between Earth and Mercury Mercury Mercury CX Sun

Which of the following answer choices correctly gives the values of sin A cos A and tan A for the triangle shown below Select the correct answer below sin A 12cos A 2 tan A 1 2 sin A 2 cos A 2 tan A 12 12 B

The circle below has center o and its radius is 6 cm Given that mAOB 30 find the length of the major arc ACB Give an exact answer in terms of and be sure to include the correct unit in your answer 30 B A Length of major arc ACB 010 cm cm cm X

Match each of the given functions with the BEST descriptor It is the only function we saw that had multiple vertical asymptotes It s domain includes all real numbers it always has a horizontal asymptote and may be increasing or decreasing It s the inverse of an exponential function It has no asymptote and normally it is increasing but slows down A It is the most basic of functions and is also a polynomial This function has two horizontal asymptotes A Linear B Rational C Root Radica D Logarithmic E Exponential F Logistic

Describe everything you can say about how the following equation has been transformed from its parent function of x f x 2 x 1 5

State two different trig equations for the following graph 2 f x 3 24 X 3T T TT 0 1 2 2 M 2 3 4 5 N a

Write a possible polynomial equation based on the graph below y 5 4 32 2 1 X 5 4 3 2 10 1 2 3 4 5 6 1 2 3 4 5

Write a polynomial in standard form that represents the area of the dhaded region x 3 X 9 X 1 X 4