Calculus Answers

6. A force of 20 lb stretches a spring from a natural length of 7 inches to a length of 12 inches. How much work was performed in stretching the spring to this length?

5. Suppose that a water tank is shaped like a right circular cone with the tip at the bottom, and has height 10 meters and radius 2 meters at the top. If the tank is full, how much work is required to pump all the water out over the top?

4. A spring has a natural length of 18 inches and a force of 201 lbs is required to stretch and hold the spring to a length of 24 inches. What is the work required to stretch the spring from a length of 21 inches to a length of 26 inches?

3. A force of F(x)=x²-cos(3x) +2, x is in meters, acts on an object. What is the work required to move the object from x=3 to x=7?

3) Solve the area bounded by the curve y = 4x-x and the lines x = -2 and y = 4.

2) Find the area bounded by the curve x = y² + 2y and the line x = 3.

1) Find the area bounded by the curve y-9-x and the x axis.

a) Horizontal Strip

b) Vertical Strip

Consider the R^2 - R function f defined by f(x,y) = (x^2 +y)/y.

Determine each of the following limits,if it exists.

a) lim (x,y) -> (0,0) f(x,y),where C1 is the curve y=x

b) lim (x,y) -> (0,0) f(x,y),Where C2 is the curve y=2x

c) lim (x,y) -> (0,0) f(x,y),Where C3 is the curve y=x^2

d) lim (x,y) -> (0,0) f(x,y)

Consider the R^2 - R function defined by f(x,y) = 3x + 2y.

Prove from first principles that lim (x,y) -> (1,-1) f(x,y) = 1

Consider the R −R2 function r r defined by   (t) = (a) Write down the domain of r (b) Is r (c) Is r continuous at t = 0? continuous at t = 2? (d) Sketch the curve r 26 . .  (t, t2) if t ∈ [−2,0] (t, t) t, t2 if if t ∈ (0,2) t ∈ [2,3]