Useful formulae of mensuration to remember :

• Volume of a cuboid $=\ell \mathrm{bh}$.

• Surface area of cuboid $=2(\ell b+b h+h \ell)$.

• Volume of cube $=\mathbf{a}^{3}$

• Surface area of cube $=6 a^{2}$

• Volume of a cone $=\frac{1}{3} \pi r^{2} h$.

• Curved surface area of cone $=\pi \mathrm{r} \ell(\ell=$ slant height $)$

• Curved surface area of a cylinder $=2 \pi \mathrm{rh}$.

• Total surface area of a cylinder $=2 \pi r h+2 \pi r^{2}$.

• Volume of a sphere $=\frac{4}{3} \pi r^{3}$.

• Surface area of a sphere $=4 \pi r^{2}$.

• Area of a circular sector $=\frac{1}{2} r^{2} \theta$, when $\theta$ is in radians.

• Volume of a prism $=$ (area of the base) $\times$ (height).

• Lateral surface area of a prism $=$ (perimeter of the base) $\times$ (height).

• Total surface area of a prism $=$ (lateral surface area $)+2$ (area of the base)

$\quad \quad \quad $ (Note that lateral surfaces of a prism are all rectangle).

• Volume of a pyramid $=\frac{1}{3}$ (area of the base) $\times$ height .

• Curved surface area of a pyramid $=\frac{1}{2}$ (perimeter of the base $) \times($ slant height) + Area of Pyramid Base . (Note that slant surfaces of a pyramid are triangles).

Some common parametric coordinates on a curve that are useful for differentiation:

• For $x^{2/3} + y^{2/3} = a^{2/3}$, take parametric coordinates $x = a \cos^3 \theta \text{and} y = a sin^3θ$

• For $\sqrt{x} + \sqrt{y} = \sqrt{a}$ take $x = a \cos^4 \theta \text{and} y = a \sin^4θ$

$\frac{x^n}{a^n} + \frac{y^n}{b^n} =1, \text{where} \quad x = a (\sinθ)^{2/n} \text{ and} \quad y = b (\cosθ)^{2/n}$

• For $c^2 (x^2 + y^2) = x^2y^2$, take x = c sec θ and y = c cosec θ

The interpretation of derivatives :

$\quad \frac{dy}{dx}$ represents the slope of the tangent.

Slope of the normal = - $\frac{dx}{dy}$

Increasing and decreasing of a function:

$\quad$ • If f’(x)>0 in x belongs to (a,b) then the function is increasing in the interval (a,b).

$\quad$ • If f’(x)<0 in x belongs to (a,b) then the function is decreasing in the interval (a,b).

Equation of tangent :

• Equation of tangent to the curve y = f(x) at $A(x_1, y_1)$ is given by $y - y_1 = (\frac{dy}{dx})_{(x_1, y_1)} (x-x_1)$

• If the tangent is parallel to the x-axis then, slope will be equal to zero.

• If the tangent is parallel to the y-axis then, slope will be equal to $\infty$.

Equation of normal:

$\quad$ The equation of the normal at $(x_1, y_1)$ to the curve $y=f(x)$ is given by the following formula:

$ (y-y_1)=m(x-x_1) \ \text{where} \ m=-\left(\frac{1}{\frac{dy}{dx}}\right) _{(x_1,y_1)} $

• Some facts regarding the normal :

• Slope of the normal drawn at point $P\left(x_1, y_1\right)$ to the curve $y=f(x)=-\left(\frac{d x}{d y}\right)_{\left(x_1, y_1\right)}$

• If the normal makes an angle of $\theta$ with the positive direction of the $x$-axis, then $-\frac{d x}{d y}=\tan \theta$ or $\frac{d y}{d x}=-\cot \theta$

• If the normal is parallel to the $x$-axis, then $\frac{d x}{d y}=0$ or $\frac{d y}{d x}=\infty$

• If the normal is parallel to the $y$-axis, then $\left(\frac{d x}{d y}\right)=\infty$ or $\frac{d y}{d x}=0$

• If the normal is equally inclined from both the axes or cuts equal intercept, then $-\left(\frac{d x}{d y}\right)= \pm 1$ or $\left(\frac{d y}{d x}\right)= \pm 1$

• The length of the perpendicular from the origin to the normal is $P^{\prime}=\left|\frac{x_1+y_1\left(\frac{d y}{d x}\right)}{\sqrt{1+\left(\frac{d y}{d x}\right)^2}}\right|$

Tangent from an external point:

$\quad$ Given a point $P(a, b)$ which does not lie on the curve $y=f(x)$, then the equation of possible tangents to the curve $y=f(x)$, passing through

$\quad$ $(a, b)$ can be found by solving for the point of contact $Q$.

$f^{\prime}(h)=\frac{f(h)-b}{h-a}$

$\quad$ And equation of tangent is $y-b=\frac{f(h)-b}{h-a}(x-a)$

Length of tangent, normal, subtangent, subnormal:

At a given point $(x_1,y_1)$

• Length of Tangent: $ L_{\text{tangent}} = y_1 \sqrt{1 + \left(\frac{dy}{dx}\right)^2} $

• Length of Normal: $ L_{\text{normal}} = \frac{y_1 \sqrt{1 + \left(\frac{dy}{dx}\right)^2}}{\left| \frac{dy}{dx} \right|} $

• Length of Subtangent: $ L_{\text{subtangent}} = \frac{y_1}{\frac{dy}{dx}} $

• Length of Subnormal: $ L_{\text{subnormal}} = y_1 \cdot \frac{dy}{dx} $

Angle between the curves:

$\quad$ Angle between two intersecting curves is defined as the acute angle between their tangents (or normals) at the point of intersection of two curves.

$ \tan \theta=\left|\frac{m_{1}-m_{2}}{1+m_{1} m_{2}}\right| $

Shortest distance between two curves :

$\quad$ Shortest distance between two non-intersecting differentiable curves is always along their common normal. (Wherever defined)

Rolle’s theorem :

$\quad$ If a function $f$ defined on $[a, b]$ is:

$\quad$ (i) continuous on $[a, b]$

$\quad$ (ii) derivable on $(a, b)$ and

$\quad$ (iii) $f(a)=f(b)$,

$\quad$ then there exists at least one real number $c$ between $a$ and $b(a<c<b)$ such that $f^{\prime}(c)=0$

Lagrange’s Mean Value Theorem (LMVT) :

$\quad$ If a function $f$ defined on $[a, b]$ is

$\quad$ (i) continuous on $[a, b]$ and

$\quad$ (ii) derivable on $(a, b)$

$\quad$ then there exists at least one real numbers between a and $b(a<c<b)$ such that $\frac{f(b)-f(a)}{b-a}=f^{\prime}(c)$

Approximation using differentiation:

The approximate value of y when the increment ∆ x is given to the independent variable x in $y = f(x)$ is:- $y + ∆y = f(x + ∆x) = f(x) + \frac{dy}{dx} ∆x\Rightarrow f(x + ∆x) = f(x) + f’(x) ∆x$

Maxima and Minima of functions of one variable :

Let $\mathrm{c}$ be a point in a domain $\mathrm{D}$ of a function $\mathrm{f}$. Then $\mathrm{f}(\mathrm{c})$ is the

• absolute maximum value of $\mathrm{f}$ on $\mathrm{D}$ if $f(c) \geq f(x)$ $\forall x \in \mathrm{D}$.
• absolute minimum value of $\mathrm{f}$ on $\mathrm{D}$ if $f(c) \leq f(x)$ $\forall x \in \mathrm{D}$.

Definition:

Let $\mathrm{c}$ be a point in a domain $D$ of a function $f$. Then $f(c)$ is the $\Rightarrow$ local maximum value of $\mathrm{f}$ if $f(c) \geq f(x)$ when $\mathrm{x}$ is near $\mathrm{c}$. $\Rightarrow$ local minimum value of $\mathrm{f}$ if $f(c) \leq f(x)$ when $\mathrm{x}$ is near $\mathrm{c}$.

Critical Point:

A critical point of a function $f$ is a point $c$ in the domain of $f$ such that either $f^{\prime}(c)=0$ or $f^{\prime}$ (c) does not exists. If $f$ has local maximum value or minimum value at $c$, then $c$ is a critical point of $f$.

The first derivative test:

Definition:

Suppose that $\mathrm{c}$ is a critical number of a continuous function $\boldsymbol{f}$.

(a) If $f$ ’ changes from positive to negative at $\mathrm{c}$, then $f$ has a local maximum at $\mathrm{c}$. (b) If $f$ ’ changes from negative to positive at $\mathrm{c}$, then $f$ has a local minimum at $\mathrm{c}$. (c) If $f^{\prime}$ does not change sign at $\mathrm{c}$ ( for example if $f^{\prime}$ is positive on both sides of $\mathrm{c}$ or negative on both sides), then $f$ has no local maximum or minimum at $\mathrm{c}$.

The Second Derivative Test:

Definition:

Suppose $f^{\prime \prime}$ is continuous near $\mathrm{c}$,

(a) If $f^{\prime}(c)=0$ and $f^{\prime \prime}(c)>0$, then fhas a local minimum at $c$. (b) If $f^{\prime}(c)=0$ and $f^{\prime \prime}(c)<0$, then $\mathrm{f}$ has a local maximum at $\mathrm{c}$.