Determine if the following series converges or diverges and explain why.

1) 1 - (4/6) + (4/6)^2 - (4/6)^3 + ...

2) 1 - (6/4) + (6/4)^2 - (6/4)^3 + ...

The temperature T in a metal ball is inversely proportional to the distance from the center of the ball, which we take to be the origin. The temperature at the point (1, 2, 2) is 120 degrees F .

a) Find the rate of change of T at (1, 1,2) in the direction towards the point (2, 1,3).

b) Show that at any point in the ball the direction of greatest increase in the temperature is given by a vector that points toward the origin.

Find the exact area of the surface obtained by rotating the curve about the x-axis.

y =

,    0 ≤ x ≤ 6

Evaluate the line integral ∫_C F⋅dr, where F(x,y,z)=−4sin(x)i+3cos(y)j−4xzk and C is given by the vector function r(t)=t^6i−t^5j+t^4k , 0≤t≤1.

part 1)

Gravel is being dumped from a conveyor belt at a rate of 40 ft3/min. It forms a pile in the shape of a right circular cone whose base diameter and height are always the same. How fast is the height of the pile increasing when the pile is 16 ft high? The height is increasing at ft/min.

part 2)

A conical water tank with vertex down has a radius of 8 feet at the top and is 11 feet high. If water flows into the tank at a rate of 30 ft3/minft3/min, how fast is the depth of the water increasing when the water is 7 feet deep?

The depth of the water is increasing at  ft/min.

part 3)

Find the partial derivatives of the function f(x,y)=−3xy+3y^3+6

fx(x,y)=

fy(x,y)=

part 4)

Find the partial derivatives of the function f(x,y)=−3x^4y^3+5

fx(x,y)=

fy(x,y)=

part 5)

Find the partial derivatives of the function f(x,y)=e^(5x−4y)

fx(x,y)=

fy(x,y)=

A rock is thrown upward from a bridge that is 31 feet above a road. The rock reaches its maximum height above the road 0.57 seconds after it is thrown and contacts the road 2.53 seconds after it was thrown.

Write a function ff that determines the rock's height above the road (in feet) in terms of the number of seconds tt since the rock was thrown.

f(t)=

part 1)

A road perpendicular to a highway leads to a farmhouse located 1 mile away. An automobile traveling on the highway passes through this intersection at a speed of 65mph. How fast is the distance between the automobile and the farmhouse increasing when the automobile is 6 miles past the intersection of the highway and the road? The distance between the automobile and the farmhouse is increasing at a rate of miles per hour.

part 2)

A boat is pulled into a dock by means of a rope attached to a pulley on the dock. The rope is attached to the front of the boat, which is 10 feet below the level of the pulley. If the rope is pulled through the pulley at a rate of 20 ft/min, at what rate will the boat be approaching the dock when 110 ft of rope is out?

The boat will be approaching the dock at  ft/min.

Hint: Sketch a diagram of this situation.

part 3)

Oil spilled from a ruptured tanker spreads in a circle whose area increases at a constant rate of 6 mi2/hrmi2/hr. How rapidly is radius of the spill increasing when the area is 10 mi2mi2?

The radius is increasing at  mi/hr.

part 4)

At noon, ship A is 40 nautical miles due west of ship B. Ship A is sailing west at 18 knots and ship B is sailing north at 15 knots. How fast (in knots) is the distance between the ships changing at 6 PM?

The distance is changing at  knots.

(Note: 1 knot is a speed of 1 nautical mile per hour.)

part 5)

A spherical balloon is inflated so that its volume is increasing at the rate of 3.8 ft3/minft3/min. How rapidly is the diameter of the balloon increasing when the diameter is 1.8 feet?

The diameter is increasing at  ft/min.

Consider the triangle which has

a=4, b=7 and c=8

Find the measure of angles ∠A=, ∠B= and ∠C= Give your answer in degrees to at least 3 decimal places.

Use the function f(x) = 2x3 - 6x2. Clearly mark your answer for each part.
Find the x and y intercepts. Give your answer as points.
State the intervals of increase and decrease.

State the local maximum and minimum points.

State the intervals of concave up and concave down.

State any in ection points.

Graph the function. Label any relevant points found in the above parts.

Find the area between the curve and the​ x-axis over the indicated interval.

y = 100 − x²​;

​[−10​,10​]

The area under the curve is ___

​(Simplify your​ answer.)

Take a circle tangent to three lines and located outside of the triangle defined by these lines. Take the three points of tangency and prove that the three corresponding Chevians pass through one point.

A bug is crawling at a velocity of f(t) = 1/1 + t meters per hour, where 0 ≤ t ≤ 1 and t is in hours.

a) Find both the left-endpoint and right-endpoint approximations for the area under the graph f(t) during this hour. Use 10-minute increments.

b) Explain what your results to Part a) mean about this bug.

A company operates two plants which manufacture the same item and whose total cost functions are C1=6.4+0.04(q1)^2 and C2=7.9+0.02(q2)^2 where q1 and q2 are the quantities produced by each plant. The company is a monopoly. The total quantity demanded, q=q1+q2, is related to the price, p, by p=40−0.02q How much should each plant produce in order to maximize the company's profit? q1=? q2=?

part 1)

Find dy/dx by implicit differentiation.

3x^6+x^5y−2xy^6=8

dy/dx=

part 4)

A campground owner has 2000 meters of fencing. She wants to enclose a rectangular field bordering a lake, with no fencing needed along the lake: see the sketch.

a) Write an expression for the length of the field: 2000-x (this is correct)

b) Find the area of the field (length times width): -x^2+2000x (this is correct)

c) Find the value of x leading to the maximum area:

d) Find the maximum area:

I got (2y^6-5yx^4-30x^5)/(x^5-12xy^5)

part 2)

sqrt (x+y)=9+x^2y^2

dy/dx= (80+4x^3y^4+36xy^2)/(1-4x^4y^3-36x^2y)

part 3)

Use implicit differentiation to find an equation of the tangent line to the curve

sin(x+y)=6x−6y at the point (π,π).

Tangent Line Equation: y= (5x/7)-(2pi/7)