1) Find the velocity and position vectors of a particle that has the given acceleration and the given initial velocity and position.

a(t) = 7i+ 4j,    v(0) = k,    r(0) = i

2) Find the tangential and normal components of the acceleration vector.

r(t) = 5(3t − t³) i + 15t² j

1. Given parametric equations below, find the values of t where the the parametric curve has a horizontal and vertical tangents.

a) x=t^2 - t, y= t^2 + t

b) x= e^(t/10)cos(t), y= e^(t/10)sin(t)

2. Find the arc length of the graph of the parametric equations on the given intervals.

a) x= 4t+2, y = 1-3t , −1 ≤ t ≤ 1

b) x= e^(t/10)cos(t), y= e^(t/10)sin(t), 0 ≤ t ≤ 2π

1. Given parametric equations below, find dy/dx , equations of the tangent and normal line at the given point.

(a) x = t^2 , y = t^3−3t at t = 1

(b) x = cos(t), y = sin(2t) at t = π/4

2. ) Given parametric equations below, find d^2y/dx^2 and determine the intervals on which the graph of the curve is concave up or concave down.

(a) x = t^2 , y = t^3−3t

(b) x = cos(t), y = sin(2t)

Find the equation of the circle ((x-a)² + (y-b)² = r² ) that goes through the points (1, 1), (0, 2), and (3, 2).  [Hint: subtract equations.]

The following DEs are not solvable as written. Perform the given variable change to turn them into something that you are equipped to solve (e.g. either linear, separable or exact) and then find the general solution. Your answer should be in the form y(x)=...y(x)=...

a) dydx=x2+y2−2xydydx=x2+y2−2xy using u=y−xu=y−x

b) xdydx−y=x2−y2−−−−−−√x>0xdydx−y=x2−y2x>0 using u=yxu=yx

c) dydx=y(xy3−1)dydx=y(xy3−1) using u=1y3

Consider the function below.

f(x) =

Find the interval(s) where the function is decreasing. (Enter your answer using interval notation. If an answer does not exist, enter DNE.)

Find the local maximum and minimum values. (If an answer does not exist, enter DNE.)

Find the inflection point. (If an answer does not exist, enter DNE.)

Consider the function below.

f(x) =

Find the interval(s) where the function is increasing. (Enter your answer using interval notation. If an answer does not exist, enter DNE.)

Find the inflection point. (If an answer does not exist, enter DNE.)

local minimum value

Solve the inequality. (Enter your answer using interval notation.)

(x + 5)²(x + 2)(x − 1) > 0

a)

Find the value of the Wronskian of the functions  f = x^7

and g = x^8 at the piont  x = 1.

b)

Let  y be the solution of the equation y ″ − 5 y ′ + 6 y = 0

satisfying the conditions y ( 0 ) = 1 and y ′ ( 0 ) = 2.

Find ln ⁡ ( y ( 1 ) ).

c)

Let y be the solution of the equation  y ″ + 2 y ′ + 2 y = 0

satisfying the conditions  y ( 0 ) = 0 and y ′ ( 0 ) = 1.

Find the value of  y at x = π.

d)

Let y be the solution of the equation  y ″ + 6 y ′ + 9 y = 0

satisfying the conditions  y ( 0 ) = 0 and y ′ ( 0 ) = 1.

Find the value of the function   f ( x ) = ln [ (y(x))/(x)] at  x = 1.

e)

One of solutions of the equation y ″ − y ′ + y = x^2 + 3x + 5

is a function of the form y = A*x^2 + B*x + C.

Find the value of the coefficient C.

f)

One of solutions of the equation y ″ − 2 y ′ + 2 y = ( x + 1 ) e^x

is a function of the form y = ( A*x + B ) e^x.

Find the value of the coefficient B.

Shane (famous Western movie name of the star character) settles down in Wyoming Territory. He lives the same distance from General Store (-78,202), the Saloon is (111, 193) and the Courthouse (202, -106). What are coordinates of Shane's home?

How far is Shane's house from the place mentioned?

What is the equation passing through the General Store, Saloon and Courthouse?

A manufacturer estimates that production​ (in hundreds of​ units) is a function of the amounts x and y of labor and capital​ used, as f(x,y)=[ 1/3x^-1/3+1/3y^-1/3]^-3. Find the number of units produced when

8 units of labor and 27 units of capital are utilized. Find and interpret

f_x (8,27) and f_y (8,27).

What would be the approximate effect on production of increasing labor by 1​ unit?

A) The number of units produced when 64 units of labor and 125 units of capital are utilized is ___

B) f_x( , ) =

c) f_y=( , )

d) The approximate effect on production of increasing labor by 1 unit would be ___ units

perform a rotation of axes with a suitable angle of rotation (no xy term) and identify the related conic.

1. x²-xy+y²=2

2. x²-3y²-8x+30y=60

A lamina occupies the region inside the circle x^2 + y^2 = 6x, but outside the circle x^2 + y^2 =9.

Find the center of mass if the density at any point is inversely proportional to its distance from the origin.

A man gets a job with a salary of $38,900 a year.
He is promised a $2,430 raise each subsequent year

During a 9-year period his total earnings are________

A lamina occupies the region inside the circle x^2 + y^2 = 6x, but outside the circle x^2 + y^2 =9.

Find the center of mass if the density at any point is inversely proportional to its distance from the origin.

In this assignment, we will explore four subspaces that are connected to a linear transformation. For the questions below, assume A is an m×n matrix with rank r, so that T(~x) = A~x is a linear transformation from R n to R m. Consider the following definitions:

• The transpose of A, denoted A^T is the n × m matrix we get from A by switching the roles of rows and columns – that is, the rows of A^T are the columns of A, and vice versa.

• The column space of A, denoted col(A), is the span of the columns of A. col(A) is a subspace of R m and is the same as the image of T.

• The row space of A, denoted row (A), is the span of the rows of A. row (A) is a subspace of Rⁿ .

• The null space of A, denoted null(A), is the subspace of Rⁿ made up of vectors x such that Ax = 0 and is equal to the kernel of T.

• The left null space of A, denoted null( A^T) , is the subspace of R^m made up of vectors y such that A^Ty = 0.

col(A), row (A), null(A), and null (A^T ) are called the four fundamental subspaces for A.

We showed in class that the pivot columns of A form a basis for col(A), and that the vectors in the general solution to Ax = 0 form a basis for null(A). Likewise, the vectors in the general solution to A^Ty = 0 form a basis for null (A^T ).

Q3: Show that row (A) and null(A) are orthogonal complements.

Q4: Show that col(A) and null ( A^T ) are orthogonal complements.

Q5: Assuming A is an m × n matrix with rank r, what are the dimensions of the four subspaces for A?