Calculus Answers

Find a vector function r(t) = <x(t),y(t),z(t)>

where r(t) is continuous everywhere except t=2

lim(t=2)r(t)= <1,0,0>

Find a vector function r(t) = <x(t),y(t),z(t)>

where r(t) is continuous everywhere except t=2

lim(t=2)r(t)= <1,0,0>

Find the volume of the solid generated by revolving the area bounded by y=4x−x², y=x, about x=3.

A private shipping company will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed 108 in. Suppose you want to mail a box with square sides so that its dimensions are   and it's girth is   What dimensions will give the box its largest volume?

lim(x,y)-(0,0) sinx/y exists.

This February, a non-profit organization will be having a bike exhibition competition in an open field that is 50 meters long. They specified that them aximum height of a single ramp is 2 meters above the ground and the minimum height is 0.5 meters below the ground. They are looking for someone who could design the entire ramp for the exhibition. The creator of the chosen design will get a chance to watch the competition for free and will receive a free bike of his/her choice.

As a bike enthusiast with knowledge on safe ramps which can be used in the competition, you decided to submit your design. Aside from the 2-dimensional illustration to be submitted, the following questions must be answered too.

1. What is the limit of a defined function in the design when it is half-way from the starting point?

2. What is the limit of a defined function in the design when it is 10 meters from the starting point?

3. At what interval/s will the design be continuous?

4. At what interval/s will the design be discontinuous?

If 𝑎 = 4𝑡^−3/2 , 𝑠 = 16 𝑤ℎ𝑒𝑛 𝑡 = 4, 𝑎𝑛𝑑 𝑠 = 25 𝑤ℎ𝑒𝑛 𝑡 = 6, find the equation of motion 𝑠 = 𝑓(𝑡) and the velocity function 𝑣(𝑡).

The points (-1,3) and (0,2) are on a curve, and at any point (x,y) on the curve 𝑑^2𝑦/𝑑𝑥^2 = 2 − 4𝑥. Find an equation of the curve.

Find the equation of the curve whose slope at any point is 𝑑𝑦/𝑑𝑥 = 𝑥^2 √𝑥 and which passes through the point (1,0).

Determine all of the points on the curve

y = 3x2 + 24x - 3 where the slope of the tangent line is 6