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Calculus
Find the area under the curve y=arccosx between x=0 and x=1 and choose the right option
a:pi
b:cos pi/2
c:2 pi
d:sin pi/2
a:pi
b:cos pi/2
c:2 pi
d:sin pi/2
Calculus
Find the limit of the function lim x->0 (sinx/x)^sinx/x-sinx
a:e^-1
b:1
c:e
d:+infinity
a:e^-1
b:1
c:e
d:+infinity
Calculus
Suppose a particle P is moving in the plane so that its coordinates are given by P(x,y), where x = 4cos2t, y = 7sin2t.
(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?[
Calculus
Sketch the graph of a continuous function f(x) satisfying the following properties: (i) the graph of f goes through the origin (ii) f0(−2) = 0 and f0(3) = 0. (iii) f0(x) > 0 on the intervals (−∞,−2) and (−2,3). (iv) f0(x) < 0 on the interval (3,∞). Label all important points.
Calculus
(a) 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.
Calculus
a) Find f'(x) using logarithmic differentiation, where f(x) = e−3x√2x−5 (6−5x)4
.
(d) Evaluate the integralZ(x3 + 1)1/3x5dx.
.
(d) Evaluate the integralZ(x3 + 1)1/3x5dx.
Calculus
(b) Differentiate the following functions with respect to x: (i) ln(1 + sin2 x) (ii) xx.
(c) Evaluate the integralZ 2x3 −4x−8 x4 −x3 + 4x2 −4x dx.
(c) Evaluate the integralZ 2x3 −4x−8 x4 −x3 + 4x2 −4x dx.
Calculus
(a) Verify that y = e2x sinx is a solution to the differential equation d2y dx2 −4
dy dx
+ 5y = 0.
(b) Differentiate the following functions with respect to x: (i) ln(1 + sin2 x) (ii) xx.
dy dx
+ 5y = 0.
(b) Differentiate the following functions with respect to x: (i) ln(1 + sin2 x) (ii) xx.
Calculus
Verify that the given family of functions solves the differential equation.
(i)
dy dt
= (1−2t)y2, y = 1 C −t + t2
.
(ii)
dy dt
= y2 sint, y = 1 C + cost
(c) Evaluate the integralZ1 0
x2 (√4−x2)3dx
(i)
dy dt
= (1−2t)y2, y = 1 C −t + t2
.
(ii)
dy dt
= y2 sint, y = 1 C + cost
(c) Evaluate the integralZ1 0
x2 (√4−x2)3dx
Calculus
Evaluate the following integrals: (i)Z xln(x + 1)dx [13 marks] (ii)Z sin3(lnx)cos2(lnx) x dx
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#340153 on Dec 2023
#340153 on Dec 2023