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Question 1.

Let f be a differentiable function such that integral(from 0 to 4) f(x)dx=8 and integral(from 0 to 4) xf'(x)dx=12. Calculate f(4).

Solution 1.

Question 2.

Calculate the area of the closed region in the xy-plane bounded by y = x-5 and y^2 =2x+5.

Solution 2.

Question 3.

Determine
(a)lim tsin(1/t)
t->0

(b)lim 3^(1/x)
x->0

©lim x ln(sinx)
x->0+

Solution 3.

Question 4.

A water trough is 10m long and a cross-section has the shape of an isosceles triangle that is 1m across at the top and is 50cm high. The trough is being filled with water at the rate of 0.4m3/min. How fast will the water level rise when the water is 40cm deep?

Solution 4.

As the trough is being filled, three values are changing with respect to time. The volume of the water in the trough, the height of the water, and the length of the base of the triangle. Let the base of the isosceles triangle be d, and height be h. Since it is isosceles, can use similar triangles to get d in terms of h:
d/h=1/0.5 so, d=2h

Volume of water in the trough,
V= area of the triangle*length of the trough
= (1/2)d*h*10
= 5*2h*h
=10h2

take derivative of V with respect to time:
dV/dt= 20h dh/dt (chain rule)
dh/dt= (1/20h) dV/dt

You know that the rate of water flow is 0.4 m3/min. This is also how fast the volume of water in the trough is changing, ie dV/dt.
You want to find out how fast the water level is rising (dh/dt) when the height is 40cm=0.4m (h)
so substitute dV/dt=0.4 and h=0.4 into dh/dt.

dh/dt= [1/(20*0.4)]*0.4= 1/20= 0.05 m/min= 5 cm/min
So, the water level is rising at 5 cm/min when the height is 40cm.

Question 5.

Given (x to infinity)(1+(1/n))^n = e
prove (x to infinity)(1+(r/n))^n = e^r

Solution 5.

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