2. A tank initially
contains 120 L of pure water. A salt mixture containing a
concentration of 1.5 g/L enters the tank at a rate of 2 L/min, and
the well-stirred mixture leaves the tank at the same rate. Find an
expression for the amount of salt in the tank at any time
*t*. Also, find the limiting amount of salt in the tank as
*t* →∞.

In: Math

Find the volume of the solid obtained by rotating the region bounded by y = x 3 , y = 1, x = 2 about the line y = −3.

Sketch the region, the solid, and a typical disk or washer (cross section in xy-plane).

Show all the work and explain thoroughly.

In: Math

Find the present and future values of a constant income stream of $3000 per year over a 13 year period at 5% annual interest compounded continuously. Round your answers to the nearest cent. In particular, ROUND your answer for the Present Value to the nearest cent BEFORE using it to compute the Future Value.

a) Present Value: $

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b) Future Value: $

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In: Math

a) Evaluate the limit lim x→0 tan(2x) / x

b) Differentiate y = x^tan(x)

c) Find the equation of the tangent line to 4x^2 + 2xy−y^2 = 4 at the point (1, 2).

d) Differentiate f(x) = arctan(x^2 + 1)

e) Differentiate f(x) = ln(cosh x)

Thank you!

In: Math

(a) Find the exact length of the curve y = 1/6 (x^{2} +
4)^{(3/2)} , 0 ≤ x ≤ 3. (b) Find the exact area of the
surface obtained by rotating the curve in part (a) about the
y-axis.

I got part a I NEED HELP on part b

In: Math

7) Determine the equations of the lines tangent to the
graph of 9?2 + 4?2 + 18? − 16? − 11 = 0 at the points where x = 0.
Where do these two lines

intersect?

In: Math

Find an equation of the tangent line to the curve cos ( x ) + 11 y ^2 = x y ^3 + 34 at the point ( 0 , √ 3 ) . Assume that y is a function of x . Express all numbers in exact form and write the equation of the tangent line in terms of x and y .

In: Math

The function F(x) = x^{2} - cos(π x) is defined on the
interval 0 ≤ x ≤ 1 radians. Explain how the Intermediate Value
Theorem shows that F(x) = 0 has a solution on the interval 0 < x
< .

In: Math

Consider a continuous, integrable, twice-differentiable function
f with input variable x.

In terms of the units of f and the units of x, choose the units of
each function or expression below:

(a) The units of f ' are

the units of

f

the units of

x

(the units of f)(the units of x

)

the units of f |

the units of x |

the units of f |

(the units of x)^{2} |

the units of f |

(the units of x)^{3} |

(b) The units of f '' are

the units of

f

the units of

x

(the units of f)(the units of x

)

the units of f |

the units of x |

the units of f |

(the units of x)^{2} |

the units of f |

(the units of x)^{3} |

(c) The units of

b | f(x)dx |

a |

are

the units of

f

the units of

x

(the units of

f

)(the units of

x

)

the units of f |

the units of x |

the units of f |

(the units of x)^{2} |

the units of f |

(the units of x)^{3} |

(d) The units of

b | f '(x)dx |

a |

are

the units of

f

the units of

x

(the units of

f

)(the units of

x

)

the units of f |

the units of x |

the units of f |

(the units of x)^{2} |

the units of f |

(the units of x)^{3} |

(e) The units of

n | f(x_{i})Δx |

i=1 |

are

the units of

f

the units of

x

(the units of

f

)(the units of

x

)

the units of f |

the units of x |

the units of f |

(the units of x)^{2} |

the units of f |

(the units of x)^{3} |

(f) The units of

f(b)−f(a) |

b−a |

are

the units of

f

the units of

x

(the units of

f

)(the units of

x

)

the units of f |

the units of x |

the units of f |

(the units of x)^{2} |

the units of f |

(the units of x)^{3} |

(g) The units of

d |

dx |

x | f(t)dt |

a |

are

the units of

f

the units of

x

(the units of

f

)(the units of

x

)

the units of f |

the units of x |

the units of f |

(the units of x)^{2} |

the units of f |

(the units of x)^{3} |

(h) The units of

d^{2} |

dx^{2} |

x | f(t)dt |

a |

are

the units of

f

the units of

x

(the units of

f

)(the units of

x

)

the units of f |

the units of x |

the units of f |

(the units of x)^{2} |

the units of f |

(the units of x)^{3} |

(i) The units of h **in**

f(a+h)−f(a) |

h |

are

the units of

f

the units of

x

(the units of

f

)(the units of

x

)

the units of f |

the units of x |

the units of f |

(the units of x)^{2} |

the units of f |

(the units of x)^{3} |

In: Math

**The principal value of tan-1(tan 3π/5) is _____________.**

(a) 2π/5

(b) -2π/5

(c) 3π/5

(d) -3π/5

In: Math

Find the area enclosed by the curves, x + y = 8 and x = y^2 − 4y + 4.

In: Math

Use the Divergence Theorem ∬SF⋅dS=∭D∇⋅FdV to find ∬SF⋅dS where F(x,y,z)=x^2i+y^2j+z^2k and S S is the surface of the solid bounded by x^2+y^2=9 , z = 0 , and z=6

In: Math

*Calc 3 multivariable question*

Find the surface area of paraboloid
z=3-2x^{2}-2y^{2}

the paraboloid lies above the xy plane

In: Math

**Partial Fractions: Problem 2**

Use the method of partial fraction decomposition to write the following rational expression as the sum of simpler rational functions whose denominators are polynomials of degree 1.

−20x+20/x^2−x−56=

In: Math

Find the solution of the Cauchy problem for the differential equationy" + 2y' + y = e–x cos x

subject to the initial conditions: y(0) = 0, y'(0) = 1.

Verify the solution obtained by direct substitution into the
equation and confirm that it satisfies the initial condition.

In: Math