The top 30 most frequently Important questions Calculus
- Test continuity and differentiability of a piecewise function.
- State and prove Leibnitz’s theorem for higher-order derivatives.
- State and prove Rolle’s theorem.
- Evaluate (\int \sin^5 x \cos^3 x dx).
- Evaluate (\int_0^\infty \frac{(\tan^{-1}x)^2}{1+x^2} dx).
- Define domain and range with examples.
- Find (\frac{dy}{dx}) for (x^y + y^x = a^b).
- Find (\frac{dy}{dx}) for (xy = \sin(x+y)).
- State and prove the Fundamental Theorem of Calculus.
- Find the area between (y^2 = 4x) and (y = 2x – 4).
- Find arc length of (y^2 = 16x) from vertex to latus rectum.
- Find maxima/minima of (f(x) = x^5 – 5x^4 + 5x^3 – 1).
- Prove (\lim_{x \to 3} \frac{x^2-9}{x-3} = 6) using ((\delta, \epsilon)) definition.
- Evaluate (\int \frac{dx}{5 + 4 \cos x}).
- Differentiate (y = x^x + \frac{1}{x^x}) w.r.t. (x).
- Find volume of solid by revolving (y^2(a+x) = x^2(a-x)) about x-axis.
- Differentiate (x^{\sin^{-1}x}) w.r.t. (\sin^{-1}x).
- Evaluate (\int \frac{dx}{(x^2+1)\sqrt{x^2+4}}).
- Evaluate (\int \frac{x^4}{1 + \sqrt{x}} dx) (definite/indefinite).
- State and prove Cauchy’s Mean Value Theorem.
- Find area of hypocycloid (x^{2/3} + y^{2/3} = a^{2/3}).
- Find perimeter of cardioid (r = a(1 + \cos \theta)).
- Evaluate (\int \tan^5 x dx).
- Evaluate (\int \frac{e^{m \tan^{-1}x}}{1+x^2} dx).
- Verify Rolle’s theorem for (f(x) = x^2 – 3x + 2) in ([1,2]).
- Evaluate (\int \sin^4 x dx).
- Evaluate (\int_0^\pi \frac{dx}{3 + 2 \sin x}).
- Define even and odd functions with examples.
- Sketch graph of a piecewise function and find domain/range.
- Evaluate (\int_0^{\pi/2} \sin^m x \cos^n x dx) using Gamma function.