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MAT1134: Differential and Integral CalculusÂļ

Chapter wise Collection of Exact Exam QuestionsÂļ

1. Functions and ModelsÂļ

  • Define Domain and Range with example.

  • Find the domain and range of:

  • \(f(x) = |x| + |x - 1| + |x - 2|\)

  • \(f(x) = 3x^2 + x + 2\)
  • \(f(x) = |x + 1| + |x| + |x - 1|\)
  • \(f(x) = \frac{x}{1 - x^2}\)
  • \(f(x) = \sqrt{x - 1} + \sqrt{5 - x}\)

2. Limits and DerivativesÂļ

  • Define limit of a function.
  • Write down the fundamental theorem on limit of a function.

  • Use L’Hospital’s Rule to evaluate the following limits:

  • \(\lim_{x \to 0} \frac{e^x - 1}{x^3}\)

  • \(\lim_{x \to 0} \frac{\ln x}{\csc x}\)
  • \(\lim_{x \to \pi/4} (1 - \tan x)\sec 2x\)
  • \(\lim_{x \to 0} (\sin x)^x\)
  • \(\lim_{x \to 0} \frac{x - \tan x}{x^3}\)
  • \(\lim_{x \to \pi/2} (\sin x)^x \tan x\)
  • \(\lim_{x \to 3} \frac{x^2 - 6x + 9}{x - 3}\)
  • \(\lim_{x \to 0} \frac{\log(1 + x)}{\sin x}\)
  • \(\lim_{x \to 0} \frac{1 - \cos x}{x^2}\)
  • \(\lim_{x \to 0^+} \frac{\ln x}{\csc x}\)
  • \(\lim_{x \to 0} \left(\frac{1}{x} - \cot x\right)\)

3. Differentiation RulesÂļ

  • From first principle, find the derivative of:

  • \(\log x^e\)

  • \(f(x) = e^x\)

  • Differentiate (including implicit differentiation):

  • \(y = \log_e\left(\frac{1 + \sin x}{1 - \sin x}\right)\)

  • \(x^y = e^{x+y}\); \(x^y = e^{x^y}\)
  • \(y = (\sin x)^x\)
  • \(y = \sin(\sqrt{1 + \cos x})\)
  • \(\frac{dy}{dx}\) from Folium of Descartes: \(x^3 + y^3 = 3xy\)
  • \(\frac{dy}{dx}\) from \(3x^2 - x^2y + 2y^3 = 0\)
  • \(\frac{dy}{dx}\) when \(x = 2\sin^{-1}\left(\frac{x}{\sqrt{1 + x^2}}\right), y = \cos^{-1}\left(\frac{1}{\sqrt{1 + x^2}}\right)\)
  • \(\frac{dy}{dx}\) of \(x^{\sin^{-1}x}\) w.r.t. \((\sin x)^x\)
  • \(\frac{d^2y}{dx^2} = \frac{9}{y^3}, \text{ if } 4x^2 - 2y^2 = 9\)
  • \(\frac{d}{dx} \tan^{-1}\left(\frac{\sqrt{1 + x^2} - 1}{x}\right)\) w.r.t. \(\tan^{-1} x\)
  • \(\frac{d}{dx} \cos^{-1}\left(\frac{1}{\sqrt{1 + x^2}}\right)\) w.r.t. \(2\sin^{-1}\left(\frac{x^2}{\sqrt{1 + x^2}}\right)\)
  • \(\cos^{-1}\left(\frac{1 - x^2}{1 + x^2}\right)\) w.r.t. \(\tan^{-1}\left(\frac{2x}{1 - x^2}\right)\)

4. Applications of DifferentiationÂļ

  • Maxima, minima, and inflection points:

  • \(f(x) = 3x^3 - 4x + 6\)

  • \(x^5 - 5x^4 + 5x^3 - 10\)
  • \(f(x) = x^3 - 9x^2 + 24x - 12\)
  • \(f(x) = x^3 - 3x^2 + 9x - 1\)

  • \(f(x) = 5x^6 - 18x^5 + 15x^4 - 10\)

  • Theorems:

  • Mean Value Theorem: Prove and find \(c\) or \(\xi\)

  • Rolle’s Theorem: State and verify

  • Others:

  • Geometrical interpretation of \(\int_a^b f(x) dx\)

5. IntegralsÂļ

(a) Define Antiderivative and Integration with example.

(b) Evaluate the following integrals (any three):
(i) \(\int \frac{e^x(1 + x)}{\cos^2 x} \, dx\)
(ii) \(\int \frac{dx}{(2x + 1)\sqrt{4x + 3}}\)
(iii) \(\int \sqrt{x^2 - 4x + 13} \, dx\)
(iv) \(\int \frac{dx}{\sqrt{x + a} \cdot \sqrt{x + b}}\)
(v) \(\int \frac{xe^x}{(x + 1)^2} \, dx\)

(c) Prove that:
\(\int e^{ax} \cos bx \, dx = \frac{e^{ax}(a \cos bx + b \sin bx)}{a^2 + b^2}\)

(d) State and prove the Fundamental Theorem of Integral Calculus.

(e) Prove that:
\(\int_{-a}^{a} f(x) dx = \begin{cases} 2 \int_0^a f(x) dx, & f(x) \text{ is even} \\ 0, & f(x) \text{ is odd} \end{cases}\)

(f) Prove that:
\(\int_0^1 \frac{\ln(1 + x)}{1 + x} \, dx = \frac{\pi}{8} \ln 2\)

(g) Obtain the reduction formula for \(\int \cos^n x dx\); hence evaluate \(\int \cos^6 x dx\)

6. Applications of IntegrationÂļ

  • Area between curves:

  • \(y^2 = 4x\) and \(y = 2x\)

  • \(x = y^2\) and \(y = x - 2\)
  • \(r = a \cos 2\theta\)

  • Area included between \(x = y^2(1 - x)\) and \(x = 1\)

  • Volume of solids of revolution:

  • About the x-axis/y-axis

  • \(y^2 = 9x\) and \(y = 3x\)
  • \(y = x^3\) between \(x = 0\) and \(x = 1\)
  • \(x^2 + y^2 = a^2\) → \(\frac{4}{3} \pi a^3\)

  • \(y^2 = 2x\) and \(y = 3x\)

  • Arc length:

  • \(y = \log \sec x\), from \(x = 0\) to \(x = \pi/3\)

  • Surface area:

  • Revolved curve: \(y = x^3\), \(x = 0\) to \(x = 1\)

7. Gamma and Beta FunctionsÂļ

  • Define Gamma and Beta functions.

  • Prove:

  • \(B(m, n) = B(n, m)\)

  • \(\Gamma\left(\frac{1}{2}\right) = \sqrt{\pi}\)
  • \(\int_0^\infty x^{1/2} e^{-x} dx = \frac{\sqrt{\pi}}{2}\)
  • \(\int_0^\infty e^{-x^2} dx = \frac{\sqrt{\pi}}{2}\)
  • Area of asteroid \(x^{2/3} + y^{2/3} = a^{2/3}\): \(\frac{3}{8}\pi a^2\)
  • Area bounded by polar curve \(r = a(1 - \cos \theta)\)

8. Euler’s Theorem and Leibnitz’s RuleÂļ

  • Euler’s Theorem:

  • Prove and verify for: \(f(x,y) = 2x^4 + 4x^2y^2 - y^4\)

  • Prove and verify for: \(f(x, y) = x^2 \log \frac{y}{x}\)

  • Leibnitz’s Theorem:

  • State and apply to find \(n^{th}\) derivative of:

    • \(y = x^3 e^{5x}\)
    • \(y = e^{\sin^{-1} x}\) at \(x = 0\)

  • Recurrence relation:

  • If \(y = a \cos(\log x) + b \sin(\log x)\), show:
    \(x^2 y_{n+2} + (2n+1)xy_{n+1} + (n^2 + 1)y_n = 0\)

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