(a) Solve the given equation.  2 cos²(theta) + sin(theta) =1

theta= ____

(b) The measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle (Enter your answers in comma separated lists).

-5pi/4

____rad

Let S be the set of vectors in R4
S ={(1,1,-1,2),(1,0,1,3),(1,-3,2,2),(0,-2-1,-2)}
(a) State whether the vectors in the set are linearly independent.
(b) Find the basis for S.
(c) Find the dimension of the column space (rank S).
(d) Find the dimension of the null space (nullity S).
(e) Find the basis for the null space of S?

Is the following map linear?

a) F(x1,x2,x3)=(0,0)

b) L:R2→R2 defined by L(x1,x2)=(3x₁−2x₂,x₂)

c) f:R→R defined by f(x)=2x

A toy tractor sold for ​$203 in 1977 and was sold again in 1989 for $ 466. Assume that the growth in the value V of the​ collector's item was exponential.

a. ) Find the value k of the exponential growth rate. Assume Vo=203

b. ) Find the exponential growth function in terms of​ t, where t is the number of years since 1977

c. ) Estimate the value of the toy tractor in 2015

d. ) What is the doubling time for the value of the toy tractor to the nearest tenth of a​ year?

e. ) Find the amount of time after which the value of the toy tractor will be​ $2931.

Let T : Pn → R be defined by T(p(x)) = the sum of all the the coefficients of p(x). Show that T is a linear transformation with dim(ker T) = n and conclude that {x − 1, x2 − 1, . . . , x^n − 1} is a basis of ker T.

The following matrix is in reduced row echelon form. Decode from the matrix the solution of the corresponding system of linear equations or state that the system is inconsistent. (If the system is dependent assign the free variable the parameter t. If the system is inconsistent, enter INCONSISTENT.)

The table shows the estimated percentage P of the population of a certain country that are mobile-phone subscribers. (End of year estimates are given.)

(a) Find the average rate of change of P.

(i) From 2003 to 2007 ........................... percentage points per year

(ii) From 2003 to 2005 ............................. percentage points per year

(iii) From 2001 to 2003 ............................... percentage points per year

(b) Estimate the instantaneous rate of growth in 2003 by taking the average of two average rates of change. (Use the average rates of change for 2001 to 2003 and 2003 to 2005.)
........................... percentage points per year

(c) Estimate the instantaneous rate of growth in 2003 by sketching a graph of P and measuring the slope of a tangent. (Sketch your graph so that it is a smooth curve through the points, and so that the tangent line has an x-intercept of 1999.3 and passing through the point

(2006, 46.6).

Round your answer to two decimal places.)
.................................... percentage points per year

Assume the reader understands derivatives, and knows the definition of instantaneous velocity (dx/dt), and knows how to calculate integrals but is struggling to understand them. Use students’ prior knowledge to provide an explanation that includes the concept and physical meaning of the integral of velocity with respect to time.

Reminder: The user is comfortable with the calculations, but is struggling with the concept. To fully address the prompt, emphasize the written explanation in English over the calculation.

Do not want hand written answer and do not copy paste. Please type. Thanks.

Write a brief account of the role of Euclid’s Fifth Axiom in the development of Geometry.

Solve by gaussian method

x-y=-1

x-z=-6

6x-2y-3z=-18

3. Solve the following system of equations.

5x- y+ z= -4

2x+ 2y-3z= -6

x-3y+ 2z= 0

Select the correct choice below:

A. There is one solution. The solution is (     ).

B. There are infinitely many solutions. The solutions (   ,z)

C. There is no solution.

4. The total number of​ restaurant-purchased meals that the average person will eat in a​ restaurant, in a​ car, or at home in a year is 150. The total number of these meals eaten in a car or at home exceeds the number eaten in a restaurant by 12. Ten more​ restaurant-purchased meals will be eaten in a restaurant than at home. Find the number of​ restaurant-purchased meals eaten in a​ restaurant, the number eaten in a​ car, and the number eaten at home.

     of the​ restaurant-purchased meals will be eaten in a restaurant.

Find the vertical and horizontal intercepts of each function.

7. f(t)= 2 (t-1) (t+2) (t-3)

12. C(t)= 4t^4+12t^3-40t^2

Find the general solution y(t) to the following ODE using ONLY the Variation of Parameters:

y"+4y'-2y = 2x²-3x+6

Section 3.3 Product and Quotient Rules and Higher-Order Derivatives

Find the derivative of the function

   g(s)=√s(s^2+8)
   g(x)=√x sin⁡x
   f(x)=x^2/(2√x+1)
   f(t)=cos⁡t/t^3
   y=sec⁡x/x
   f(x)=sin⁡x cos⁡x
   y=(2e^x)/(x^2+1)


Find equation of the tangent line to the graph of the function f(x)=(x+3)/(x-3) at the point (4, 7)
Find the equation of the tangent line to the graph of the function ??=24?3 at the point (1, 2).

Kevin is organizing luxury bus tickets to Lynnwood from Shoreline. If the ticket price is $200, then Kevin can sell 80 tickets. For each five dollars Kevin increases the price, one fewer ticket is sold.

(a) If tickets are sold for $(200 + 5x) each (that is, if Kevin raises the price by $5 x times), how many are sold? _____ tickets

(b) Express the total amount of money Kevin gets by selling tickets for $(200 + 5x) each. Simplify your answer. amount=_____

(c) What price should the ticket be for Kevin to get the most money? price= ________