Vardaan Institute - Detailed Solutions: Master Sheet (Basic Mathematics)

Section A: Differentiation & Rates of Change Solutions

Differentiate $y = 3x^4 - 5x^2 + 7x - 9$ with respect to $x$.

Sol: $\frac{dy}{dx} = 3(4x^3) - 5(2x) + 7 = 12x^3 - 10x + 7$.

If position of a particle is $x(t) = 2t^3 - 9t^2 + 12t + 5$, find the velocity at $t = 2\text{ s}$ and acceleration when velocity is zero.

Sol: $v(t) = \frac{dx}{dt} = 6t^2 - 18t + 12$. At $t=2\text{ s}$, $v(2) = 6(4) - 18(2) + 12 = 0\text{ m/s}$. Acceleration $a(t) = 12t - 18$. At $v=0$, $t=1$ or $t=2$. At $t=1$, $a = -6\text{ m/s}^2$; at $t=2$, $a = 6\text{ m/s}^2$.

Evaluate $\frac{d}{dx} \left( \frac{\sin x}{x} \right)$ using quotient rule.

Sol: Using $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v u' - u v'}{v^2}$: $\frac{x \cos x - \sin x(1)}{x^2} = \frac{x \cos x - \sin x}{x^2}$.

Section B: Integration & Definite Integrals Solutions

Evaluate the indefinite integral $\int (4x^3 - 6x^2 + 2x - 5) \, dx$.

Sol: $\int (4x^3 - 6x^2 + 2x - 5) dx = 4\left(\frac{x^4}{4}\right) - 6\left(\frac{x^3}{3}\right) + 2\left(\frac{x^2}{2}\right) - 5x + C = x^4 - 2x^3 + x^2 - 5x + C$.

Calculate the work done $W = \int_{1}^{3} (3x^2 + 2x) \, dx$.

Sol: $\left[ x^3 + x^2 \right]_1^3 = (3^3 + 3^2) - (1^3 + 1^2) = (27 + 9) - (1 + 1) = 36 - 2 = 34\text{ J}$.

Section C: Trigonometry & Maxima/Minima Solutions

Find the maximum and minimum values of $y = 3\sin x + 4\cos x$.

Sol: Expression of form $a\sin x + b\cos x$ has max $\sqrt{a^2+b^2} = \sqrt{3^2+4^2} = 5$ and min $-\sqrt{a^2+b^2} = -5$.

Find the critical points of $f(x) = x^3 - 3x + 2$ and determine where it achieves relative maxima and minima.

Sol: $f'(x) = 3x^2 - 3 = 0 \implies x = \pm 1$. $f''(x) = 6x$. At $x = -1$, $f''(-1) = -6 < 0$ (Maximum); at $x = 1$, $f''(1) = 6 > 0$ (Minimum).

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