Find the exact volume of the solid obtained by rotating the region { (x, y) : 4 ≤ x ≤ 6; √8x ≤ y ≤ x^2 } about the line x = 2.
In: Math
Use the product rule or quotient rule, as appropriate, to compute each of the following derivatives:
A. sin^2(x)
B. sin(x) cos(x)
C. sec(x) tan(x)
D. x^2ln(x)
E. xtan(x)
In: Math
A ball of radius 17 has a round hole of radius 7 drilled through its center. Find the volume of the resulting solid.
In: Math
Approximate the arc length of the curve over the interval using Simpson’s Rule SN with ?=8. ?=2?^(−?^2), on ?∈[0,2] (Use decimal notation. Give your answer to four decimal places.) ?8≈
In: Math
Show that this function has a positive maximum on the interval 0<_x_<1:
f(x)=x(x-1)e^x
In: Math
In the following problems provide a complete explanation of why it is true(1) Letnbe an integer. Justify the following statements.(a) The last digit ofnis even if and only ifnis divisible by 2.(b) The last two digits ofnare divisible by 4 if and only ifnis divisible by 4.(c) The last three digits ofnare divisible by 8 if and only ifnis divisible by 8.(d) The lastkdigits ofnare divisible by 2kif and only ifnis divisible by 2k.(2) Letnbe an integer. Justify the following statements.(a) The integernis divisible by 3 if and only if the sum of the digits is divisible by 3.(b) The integernis divisible by 9 if and only if the sum of the digits is divisible by 9.(3) What is the last nonzero digit at the end of 10! ? What is the last nonzero digit at the endof 100! ? What is the last nonzero digit at the end of 1,000,000! ? Describe a procedure forfinding the last non-zero digit at the end ofn! for anyn. Use that procedure to find thelast non-zero digit for the factorial of your student id number
In: Math
Consider the tetrahedron with vertices A= (0;1;1); B= (4;0;1); C=(2;1;3), and D= (2;2;0). a) What is the area of the parallelogram spanned by vector AB and vector AD? b) Determine the angle between the two vectors AB and CD. c) Find an equation of plane through vector BC and point D.
In: Math
Find the largest volume V = xyz of a box for which the point (x, y, z) is on the ellipsoid x^2 + 4y^2 + 16z^2 = 48
In: Math
Do a news search on coronavirus, specifically the mathematical modeling of its spread. Explain how the mathematics we are studying applies to this modeling.
In: Math
Use Lagrange multipliers to find the point on the plane 2x + y + 2z = 6 which is closest to the origin
In: Math
1a.Find the equation of the tangent plane to the surface √ x + √y + √ z = 4 at P(1, 1, 4).
1b.Let f be a function of x and y such that fx = 3x − 5y and fy = 2y − 5x, which of the following is always TRUE?
a. (0, 0) is not a critical point of f.
b. f has a local minimum at (0, 0)
c. f has a local maximum at (0, 0)
d. (0, 0, f(0, 0)) is a saddle point of z = f(x, y).
e. None of these.
In: Math
Verify the Divergence Theorem for the vector field and region: ?=〈9?,3?,8?〉 and the region ?2+?2≤1, 0≤?≤8 ∬s F * ds = ∭r div(?)??=
In: Math
(1 point) The temperature at a point (x,y,z) is given by ?(?,?,?)=200?−?2−?2/4−?2/9, where ? is measured in degrees Celsius and x,y, and z in meters. There are lots of places to make silly errors in this problem; just try to keep track of what needs to be a unit vector. Find the rate of change of the temperature at the point (1, 1, 1) in the direction toward the point (-1, -1, -1).
In which direction (unit vector) does the temperature increase the fastest at (1, 1, 1)?
What is the maximum rate of increase of ? at (1, 1, 1)?
In: Math
A tank contains 2340 L of pure water. A solution that contains
0.07 kg of sugar per liter enters a tank at the rate 9 L/min The
solution is mixed and drains from the tank at the same rate.
(a) How much sugar is in the tank initially?
(kg)
(b) Find the amount of sugar in the tank after ?t minutes.
amount = (kg)
(your answer should be a function of ?t)
(c) Find the concentration of sugar in the solution in the tank
after 8787 minutes.
concentration = (kg/L)
In: Math
II. Free Response. Please show all work to receive full credit. Take the derivatives of the following functions, using any method.
a. If f(x)=8-(6)/(\sqrt(x))+(10)/(x^(2)) , find f'(x).
b. If g(x)=(2-x)^(79), find g'(x).
c. If h(x)=(x^(3)+7x-1)(5x+2), find h'(x).
d. If j(x)=(7)/(x^(2))+(10)/(x^(6)), find j'(x).
e. If k(x)=\sqrt(x)(x^(2)-(2)/(x)) , find k'(x).
In: Math