Answer any ten of the following questions (2 marks each)
. All questions solved below.Define limit of a function at x = a.
A function f(x) is said to have a limit L as x approaches a if for every epsilon > 0, there exists a delta > 0 such that |f(x) - L| < epsilon whenever 0 < |x - a| < delta.
Check the continuity of f(x) = {x^2 when x != 1, 2 when x = 1}.
Limit (x->1) f(x) = Limit (x->1) x^2 = 1
. However, the functional value f(1) = 2. Since the limit (1) is not equal to the functional value (2), the function is discontinuous at x = 1.Use definition to find the derivative of f(x) = sqrt(x), x > 0.
f'(x) = lim(h->0) [sqrt(x+h) - sqrt(x)] / h
. Rationalizing the numerator: lim(h->0) [(x+h) - x] / [h(sqrt(x+h) + sqrt(x))] = lim(h->0) 1 / (sqrt(x+h) + sqrt(x)) = 1 / (2*sqrt(x)).Write the geometrical interpretation of Rolle's theorem along with a diagram.
Rolle's theorem states that if a function is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists at least one point c in (a, b) where the tangent is horizontal (slope = 0)
.Find values of x for local maxima or minima: f(x) = 2x^3 - 21x^2 + 36x - 20.
f'(x) = 6x^2 - 42x + 36
. Setting f'(x) = 0: x^2 - 7x + 6 = 0 => (x-6)(x-1) = 0. Critical points are x = 1 and x = 6.Evaluate: Limit (x->0) [e^x - e^(-x) - 2x] / [x - sin x].
Using L'Hopital's Rule repeatedly: lim [e^x + e^(-x) - 2] / [1 - cos x] = lim [e^x - e^(-x)] / [sin x] = lim [e^x + e^(-x)] / [cos x] = (1 + 1) / 1 = 2.
Find fx and fy for f(x, y) = tan^-1(y/x).
Show that Integral(0 to a) f(x) dx = Integral(0 to a) f(a-x) dx.
Let x = a - t, then dx = -dt
. When x=0, t=a; when x=a, t=0. The integral becomes Integral(a to 0) f(a-t) (-dt) = Integral(0 to a) f(a-t) dt. Replacing t with x gives the result.If f(x) is an odd function, show that Integral(-a to a) f(x) dx = 0.
Integral(-a to a) f(x) dx = Integral(-a to 0) f(x) dx + Integral(0 to a) f(x) dx
. Letting x = -t in the first part and using f(-t) = -f(t) (odd function property), the terms cancel out to 0.Formula for volume of solid of revolution about X-axis.
Volume V = Integral(x1 to x2) pi * [f(x)]^2 dx.
Answer any five of the following questions (10 marks each)
. All questions solved below.(a) Use definition to show Limit (x->2) (3x - 4) = 2.
For any epsilon > 0, we need |(3x - 4) - 2| < epsilon. This implies |3x - 6| < epsilon or 3|x - 2| < epsilon. Thus |x - 2| < epsilon/3. Choosing delta = epsilon/3 proves the limit.
(b) If y = tan^-1(x), prove (1+x^2)y1 = 1 and the nth derivative relation. Find (yn)0.
y1 = 1 / (1+x^2) => (1+x^2)y1 = 1
. Differentiating n times using Leibnitz's theorem: (1+x^2)yn+1 + n(2x)yn + [n(n-1)/2](2)yn-1 = 0. At x=0, (yn+1)0 = -n(n-1)(yn-1)0.(a) Show differentiability implies continuity. Give example of continuous but not differentiable.
If f is differentiable at a, lim [f(x)-f(a)] = lim [(f(x)-f(a))/(x-a)] * (x-a) = f'(a) * 0 = 0, so lim f(x) = f(a) (continuous)
. Example: f(x) = |x| at x=0 is continuous but has no unique tangent.(b) State and prove Leibnitz's theorem.
The theorem provides a formula for the nth derivative of the product of two functions: (uv)n = u_n*v + nC1*u_n-1*v1 + ... + u*v_n
.(a) State and prove Lagrange's Mean-Value Theorem.
If f is continuous on [a, b] and differentiable on (a, b), there exists c in (a, b) such that f'(c) = [f(b) - f(a)] / (b - a)
.(b) Derive expansion of sin(x) in powers of x.
Using Maclaurin's series: f(x) = f(0) + xf'(0) + (x^2/2!)f''(0) + ...
. For f(x) = sin x: f(0)=0, f'(0)=1, f''(0)=0, f'''(0)=-1. Expansion: x - x^3/3! + x^5/5! - ....(a) Show the largest rectangle with a given perimeter is a square.
Perimeter P = 2(x+y) => y = P/2 - x. Area A = x*y = x(P/2 - x) = (Px/2) - x^2. dA/dx = P/2 - 2x. Setting to 0, x = P/4. Then y = P/2 - P/4 = P/4. Since x = y, it is a square.
(b) If u = log(x^2 + y^2), show d2u/dx2 + d2u/dy2 = 0. Solve Euler theorem problem.
Partial derivatives: du/dx = 2x/(x^2+y^2), d2u/dx2 = [2(y^2-x^2)]/(x^2+y^2)^2. Similarly d2u/dy2 = [2(x^2-y^2)]/(x^2+y^2)^2. Sum = 0. For Euler's: z = tan u is homogeneous of degree 2, so x*du/dx + y*du/dy = n*f(u)/f'(u) = 2*tan u / sec^2 u = 2 sin u cos u = sin 2u.
(a) Find perimeter of circle x^2 + y^2 = a^2 using integration.
y = sqrt(a^2 - x^2). dy/dx = -x/sqrt(a^2 - x^2). Length = 4 * Integral(0 to a) sqrt(1 + (dy/dx)^2) dx = 4 * Integral(0 to a) [a / sqrt(a^2 - x^2)] dx = 4a [sin^-1(x/a)](0 to a) = 4a * (pi/2) = 2*pi*a.
(b) Volume and Surface Area of cycloid x=a(theta+sin theta), y=a(1+cos theta).
Volume V = Integral pi * y^2 dx
. For cycloid revolving about x-axis: V = 5*pi^2*a^3. Surface Area S = Integral 2*pi*y ds = 64*pi*a^2 / 3.Would you like me to solve the reduction formulas for sin^n(x) or sin^m(x)cos^n(x) from Section B in step-by-step detail?