1) Evaluate the integral from 0 to 1 (e^(2x) (x^2 + 4) dx)
(a) What is the first step of your ‘new’ integral?
(b) What is the final antiderivative step before evaluating?
(c) What is the answer in simplified exact form?

2) indefinite integral (cos^2 2theta) / (cos^2 theta) dtheta
(a) What is the first step of your ‘new’ integral?
(b) What is the simplified integral before taking the antiderivative?
(c) What is the answer in simplified form?

A mass of 1.5 kg stretches a spring 0.05 mm. The mass is in a medium that exerts a viscous resistance of 240 NNwhen the mass has a velocity of 6 msms. The viscous resistance is proportional to the speed of the object.

Suppose the object is displaced an additional 0.06 mm and released.

Find an function to express the object's displacement from the spring's natural position, in mm after tt seconds. Let positive displacements indicate a stretched spring, and use 9.8 ms2ms2 as the acceleration due to gravity.

Find the directional derivative of f at the given point in the direction indicated by the angle θ.

f(x, y) = y cos(xy),    (0, 1),    θ = π/4

D_uf(0, 1) =

Find all distinct roots (real or complex) of z2+(−6+i)z+(25+15i). Enter the roots as a comma-separated list of values of the form a+bi. Use the square root symbol '√' where needed to give an exact value for your answer. z = ???

Given that v⃗ 1=[−3,1] and v⃗ 2=[2,−1] are eigenvectors of the matrix

determine the corresponding eigenvalues. λ1= . λ2= .

Given the curve C in parametric form :

C : x = 2cos t , y = 2sin t , z = 2t ; 0≤ t ≤ 2pi

a) the velocity v(t)

b) the speed ds/dt

c) the acceleration a(t)

d) the unit tangent vector T(t)

e) The curvature k and the normal vector N(t)

f) the binormal vector B(t)

g) The tangential and normal components of accelertation

Questions Determine whether or not the vector field is conservative. If it is conservative, find a vector f f such that . F→=∇f. → F ( x , y , z ) =< y cos x y, x cos x y , − sin z > F→ conservative. A potential function for → F F→ is f ( x , y , z ) = f(x,y,z)= + K. (Type "DNE" if → F F→ is not conservative.)

Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.)

dx,    n = 4

(a) the Trapezoidal Rule


(b) the Midpoint Rule


(c) Simpson's Rule

Find all distinct (real or complex) eigenvalues of A. Then find the basic eigenvectors of A corresponding to each eigenvalue.
For each eigenvalue, specify the number of basic eigenvectors corresponding to that eigenvalue, then enter the eigenvalue followed by the basic eigenvectors corresponding to that eigenvalue.

A = 11 −10

17 −15

Number of distinct eigenvalues: ?

Number of Vectors: ?

? : {???}

Explain in your own words how you approach solving any quadratic equation by factoring. What are the key things you keep in mind as you’re working towards finding the answer(s)? Your explanation must be at least a few sentences long. Remember to outline your specific, step-by-step process.

Solve the following initial-value problem:

(ye^2xy + x)dx − (y² − xe^2xy)dy = 0, y(2) = 0

Find the equation of the osculating circle at the local minimum of

3x^3 -7x^2 +(11/3)x+4

Revenue, cost, and profit. The price–demand equation and the cost function for the production of table saws are given, respectively, by

x=6,000−30pandC(x)=72,000+60xx=6,000−30pandC(x)=72,000+60x

where x is the number of saws that can be sold at a price of $p per saw and C(x) is the total cost (in dollars) of producing x saws.

• (F) Graph the cost function and the revenue function on the same coordinate system for 0≤x≤6,000. Find the break-even points, and indicate regions of loss and profit.

• (G) Find the profit function in terms of x.

• (H) Find the marginal profit.

• (I) Find P'(1,500) and P′(3,000) and interpret these quantities.

• Please write the answer clear Thank you!!

Consider the function f(x)= tan 1/x

Use the following table to guess the limit f(x) as x goes to 0+