Calculus Answers

Verify that the given family of functions solves the differential equation.
(i) dy⁄dx= (1-2t)y², y=⅟(C – t + t²)
(ii) dy⁄dx= y² sin t, y=⅟(C + cos t)

(b) A wire of length 100 centimeters is cut into two pieces. One piece is bent to form a square.The other piece is bent to form an equilateral triangle. Find the dimensions of the two pieces of wire so that the sum of the areas of the square and the triangle is minimized.

a) A bowl of water has a temperature of 50◦C. It is put into a refrigerator where the temperature is 5◦C. After 0.5 hours, the water is stirred and its temperature is measured to be 20◦C. It is then left to cool further, take t in Newton’s law of cooling to be measured in minutes. Use Newton’s law of cooling to predict when the temperature will be 10◦C
b)) Find the volume of the solid generated by revolving the region bounded by the curves y = x^2 and y = 4x − x^2 about the line y = 6

4. (a) Suppose a particle P is moving in the plane so that its coordinates are given by P(x, y),
where x = 4 cos 2t, y = 7 sin 2t.
(i) By finding a, b ∈ R such that x2 a2 + y2b2
= 1, show that P is travelling on an elliptical
path. [10 marks]
(ii) Let L(t) be the distance from P to the origin. Obtain an expression for L(t).[8 marks]
(iii) How fast is the distance between P and the origin changing when t = π/8?[7 marks]
(b) A wire of length 100 centimeters is cut into two pieces. One piece is bent to form a square.
The other piece is bent to form an equilateral triangle. Find the dimensions of the two
pieces of wire so that the sum of the areas of the square and the triangle is minimized.

2. (a) A bowl of water has a temperature of 50
◦
C. It is put into a refrigerator where the temper
ature is 5
◦
C. After 0.5 hours, the water is stirred and its temperature is measured to be
20
◦
C. It is then left to cool further, take t in Newton’s law of cooling to be measured in
minutes. Use Newton’s law of cooling to predict when the temperature will be 10
◦
C.
[15 marks]
(b) Verify that the given family of functions solves the difffferential equation.
(i)
dy
dt
= (1 2t)y
2
, y =
1
C t + t2 . [10 marks]
(ii)
dy
dt
= y
2
sin t, y =
1
C + cost
. [10 marks]
(c) Evaluate the integral
Z
1
0
x
2
(√4 x2
)
3 dx [15 marks]

Find the solution of the initial-value problem and determine its interval of existence (i.e domain of the resulting function y).

(i)dy/dt=(1+t²)/t , y(t = 1) = 0

(ii) (t + 1) dy/dt = 1- y , y(t = 0) = 3 [Hint: Let A = -(±e^(-c))]

3. (a) Find the volume of the solid generated by revolving the region bounded by the curves y=x^2 and y=4x−x^2 about the line y=6.

(b) Sketch the graph of a continuous function f(x) satisfying the following properties:
(i) the graph of f goes through the origin
(ii) f′(−2) = 0 and f′(3) = 0.
(iii) f′(x) > 0 on the intervals (−∞,−2) and (−2,3).
(iv) f′(x) < 0 on the interval (3,∞).
Label all important points.

(b) A wire of length 100 centimeters is cut into two pieces. One piece is bent to form a square. The other piece is bent to form an equilateral triangle. Find the dimensions of the two pieces of wire so that the sum of the areas of the square and the triangle is minimized.

Find the volume of the solid generated by revolving the region bounded by the curves y =x2 and y =4x−x2 about the line y = 6.

(b) Sketch the graph of a continuous function f(x) satisfying the following properties: (i) the graph of f goes through the origin
(ii) f(−2) = 0 and f(3) = 0.
(iii) f(x) > 0 on the intervals (−∞,−2) and (−2,3).
(iv) f(x) < 0 on the interval (3,∞). Label all important points.

Differentiate the following functions with respect to x: (i) ln(1 + sin2 x) (ii) xx.
(c) Evaluate the integral 2x3 −4x−8 /x4 −x3 +4x2 −4x dx.