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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 + y²/b² = 1, show that P is travelling on an elliptical
path.
(ii) Let L(t) be the distance from P to the origin. Obtain an expression for L(t).
(iii) How fast is the distance between P and the origin changing when t = π/8?
(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.
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) f¹(-2) = 0 and f¹(3) = 0.
(iii) f¹(x) > 0 on the intervals (-∞, -2) and (-2; 3).
(iv) f¹(x) < 0 on the interval (3;∞).
Label all important points.
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 ϵ R such that x²/a² + y²/b² =1, show that P is travelling on an elliptical path.
(ii) Let L(t) be the distance from P to the origin. Obtain an expression for L(t).
(iii) How fast is the distance between P and the origin changing when t = π/8?
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.
(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) Find the integral of[ (x^3+1)^1/3x
^5]dx
(b) 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.
(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 x^x
(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).]
(b) Evaluate the following integrals: (i) xln(x +1)dx (ii) sin2(ln x)cos2(lnx)/ x dx.
