Application of Derivatives

Application of Derivatives

Derivatives are used to find the rate of change of a quantity. Derivatives have a lot of applications like calculating the velocity, predicting sales over time, predicting bacterial growth, the spread of infection, etc.

The application of derivatives is mainly around the concepts of rate of change of quantities, approximation, tangent and normal to a curve, increasing and decreasing function and minima and maxima of the function. Before, looking in details about this topic, let us recall what is a derivative.

Rate of Change

The value obtained after differentiating a function is called the derivative.

If two related quantities are changing over time, the rates at which the quantities change are related. For example, consider a balloon with air, both the radius and the volume of the balloon increase as the air in it increases.

If a variable quantity $y$ depend on and vary with a quantity $x$, then the rate of change of $y$ with $x$ is $\frac{d y}{d x}$.

For example, the rate of change of displacement ($s$) of an object w.r.t. time ($t$) is velocity $(\mathrm{v})$.

$v=\frac{d s}{d t}$

Let us consider the stone being thrown at a lake causing circular waves at a speed of 4cm per second. Now let us find how fast the enclosed area increases.

The area of a circle with radius $r$ is given by $\mathrm{A}=\pi r^2$. Therefore, the rate of change of area with respect to time $t$ is

$
\frac{d \mathrm{~A}}{d t}=\frac{d}{d t}\left(\pi r^2\right)=\frac{d}{d r}\left(\pi r^2\right) \cdot \frac{d r}{d t}=2 \pi r \frac{d r}{d t}
$

As the waves move at a speed of 4cm,

$
\frac{d r}{d t}=4 \mathrm{~cm} / \mathrm{s}
$

$\frac{d \mathrm{~A}}{d t}=2 \pi(10)(4)=80 \pi$

Thus, the enclosed area is increasing at the rate of $80 \pi \mathrm{~cm}^2 / \mathrm{s}$, when $r=10 \mathrm{~cm}$.

Linear Approximation

Linear approximation is used to estimate the value of the function close to the chosen point.

Let $f:(a, b) \rightarrow \mathbb{R}$ be a differentiable function and $x_0 \in(a, b)$. We define the linear approximation $L$ of $f$ at $x_0$ by

$
L(x)=f\left(x_0\right)+f^{\prime}\left(x_0\right)\left(x-x_0\right), \quad \forall x \in(a, b)
$

For instance, Let us assume that the shape of a soap bubble is a sphere. Now, let us approximate the increase in the surface area of a soap bubble as its radius increases from 5 cm to 5.2 cm.

The surface area of the soap bubble with radius $r$ is $S(r)=4 \pi r^2$.

$S'(r) = 8 \pi r$

Change in the surface area

$
\begin{aligned}
S(5.2)-S(5) & \approx S^{\prime}(5)(5.2-5) \\
& =8 \pi(5)(0.2)=8 \pi \mathrm{~cm}^2
\end{aligned}
$

Now, let us look into the types of error.

Absolute Error

$\Delta \mathrm{x}$ or $d x$ is called absolute error in $x$.

Relative Error

$\frac{\Delta \mathrm{x}}{\mathrm{x}}$ or $\frac{d x}{\mathrm{x}}$ is called the relative error in $x$

Percentage Error

$\frac{\Delta \mathrm{x}}{\mathrm{x}} \cdot 100$ or $\frac{d x}{\mathrm{x}} \cdot 100$ is called the percentage error in $x$.

Tangent and Normal to a Curve

A tangent is a line touching the curve at only one point without passing through it. The tangent to a curve at a point P on it is defined as the limiting position of the secant $P Q$ as the point $Q$ approaches the point $P$ provided such a limiting position exists. The slope and equation of the tangent can be found with the use of derivatives if the curve of the function is given.

Let $P\left(x_0, y_0\right)$ be a point on the continuous curve $y=f(x)$, then the slope of the tangent to the curve at point $P$ is

$
\begin{aligned}
& \left(\frac{d y}{d x}\right)_{\left(x_0, y_0\right)} \\
& \Rightarrow\left(\frac{d y}{d x}\right)_{\left(x_0, y_0\right)}=\tan \theta=\text { slope of tangent at } P
\end{aligned}
$
Where $\theta$ is the angle which the tangent at $\mathrm{P}\left(\mathrm{x}_0, \mathrm{y}_0\right)$ makes with the positive direction of the $x$-axis as shown in the figure.

• If the tangent is parallel to $x$-axis then $\theta=0^{\circ}$.

  $
  \begin{aligned}
  & \Rightarrow \quad \tan \theta=0 \\
  & \therefore \quad\left(\frac{d y}{d x}\right)_{\left(x_0, y_0\right)}=0
  \end{aligned}
  $

• If the tangent is perpendicular to $x$-axis then $\Theta=90^{\circ}$

  $
  \begin{aligned}
  & \Rightarrow \quad \tan \theta \rightarrow \infty \quad \text { or } \quad \cot \theta=0 \\
  & \therefore\left(\frac{d x}{d y}\right)_{\left(x_0, y_0\right)}=0
  \end{aligned}
  $

Equation of Tangent

Let the equation of curve be $y=f(x)$ and let point $P\left(x_0, y_0\right)$ lies on this curve.

The slope of the tangent to the curve at a point $P$ is$
\left(\frac{d y}{d x}\right)_{\left(x_0, y_0\right)}
$

Hence, the equation of the tangent at point $P$ is $
\left(y-y_0\right)=\left(\frac{d y}{d x}\right)_{\left(x 0, y_0\right)} \cdot\left(x-x_0\right)
$

Equation of Normal

The equation of normal of a curve is $\left(y-y_0\right) f^{\prime}\left(x_0\right)+\left(x-x_0\right)=0$.

Monotonicity (Increasing and decreasing functions)

Monotonicity is the behaviour of the function whether it is increasing or decreasing. A function is said to be monotonic if it is either increasing or decreasing in its entire domain. The behaviour of the function is determined using the concept of derivatives.

Increasing and Decreasing Functions: Application of Derivatives Class 12

A function is said to be an increasing function if it increases throughout its domain. A function $f(x)$ is increasing in the interval $[a, b]$ if $f(x_2) \geq f(x_1)$ for all $x_2 > x_1$, where $x_1, x_2 \in [a, b]$. If the function is differentiable, then $\frac{d}{dx}f(x) \geq 0 \quad \forall x \in (a, b)$ represents an increasing function, while $\frac{d}{dx}f(x) > 0 \quad \forall x \in (a, b)$ represents a strictly increasing function.

A function is said to be a decreasing function if it decreases throughout its domain. A function $f(x)$ is decreasing in the interval $[a, b]$ if $f(x_2) \leq f(x_1)$ for all $x_2 > x_1$, where $x_1, x_2 \in [a, b]$. If the function is differentiable, then $\frac{d}{dx}f(x) \leq 0 \quad \forall x \in (a, b)$ represents a decreasing function, while $\frac{d}{dx}f(x) < 0 \quad \forall x \in (a, b)$ represents a strictly decreasing function.

Examples of Increasing and Decreasing Functions

For example, consider the function $f(x) = x$. Its derivative $f'(x) = 1 > 0$, which means $f(x)$ is an increasing function. Similarly, for $f(x) = -x$, we have $f'(x) = -1 < 0$, making it a decreasing function.

Application of Derivatives in Monotonicity

One of the main applications of derivatives class 12 is the study of monotonicity. By analyzing the derivative of a function, we can determine where the function is increasing or decreasing. This has practical real-life applications, such as finding the increase or decrease in profit, sales, production, or cost. Understanding these trends is crucial in fields like economics, business, and engineering.

Behaviour of function due toMonotonicity

• $f'(x) > 0 \implies$ Function is strictly increasing

• $f'(x) \ge 0 \implies$ Function is increasing

• $f'(x) < 0 \implies$ Function is strictly decreasing

• $f'(x) \le 0 \implies$ Function is decreasing

Using these application of derivatives class 12 solutions and formulas, students can solve various problems related to real-world scenarios involving growth and decline effectively.

Maxima and Minima

The maximum and minimum values of a curve can be determined using derivatives.

Let $f$ be a function defined on an open interval $I$. Let $f$ be continuous at a critical point $c$ in $I$. Then

(i) If $f^{\prime}(x)$ changes sign from positive to negative as $x$ increases through $c$, i.e., if $f^{\prime}(x)>0$ at every point sufficiently close to and to the left of $c$, and $f^{\prime}(x)<0$ at every point sufficiently close to and to the right of $c$, then $c$ is a point of local maxima.

(ii) If $f^{\prime}(x)$ changes sign from negative to positive as $x$ increases through $c$, i.e., if $f^{\prime}(x)<0$ at every point sufficiently close to and to the left of $c$, and $f^{\prime}(x)>0$ at every point sufficiently close to and to the right of $c$, then $c$ is a point of local minima.

Let $y=f(x)$ be a real function defined at $x=a$. Then the function $f(x)$ is said to have a maximum value at $x=a$ if $f(x) \leq f(a) \quad \forall a \in$ R .

And also the function $f(x)$ is said to have a minimum value at $x= a$, if $f(x) \geq f(a) \quad \forall a \in R$

If at $x=a$ the shape of the curve changes from increasing to decreasing or from decreasing to increasing. Then $\mathrm{x}=\mathrm{a}$ is known as the point of inflection and $f^{\prime \prime}(x)=0$ at $x=a$

These maximum and minimum values has a lot of applications in real-life problems. It can be used to determine the maximum profit, to determine the dosage of a medicine, etc.

Now, let us summarize and recall the application of derivatives formulas.

Application of Derivatives Formulas

The application of derivatives formulas includes the formulas of rate of change, approximation and errors, equation of tangent and normal to the curve, conditions for increasing and decreasing functions, and maxima and minima.

Rate of Change

If two related quantities are changing over time, the rates at which the quantities change are related. If a variable quantity $y$ depend on and vary with a quantity $x$, then the rate of change of $y$ with $x$ is $\frac{d y}{d x} = \frac{\Delta y}{\Delta x}$.

Linear Approximation and Error

Linear approximation is used to estimate the value of the function close to the chosen point.

Let $f:(a, b) \rightarrow \mathbb{R}$ be a differentiable function and $x_0 \in(a, b)$. We define the linear approximation $L$ of $f$ at $x_0$ by

$
L(x)=f\left(x_0\right)+f^{\prime}\left(x_0\right)\left(x-x_0\right), \quad \forall x \in(a, b)
$

Tangent and Normal to a Curve

The equation of tangent to a curve at point $P$ is $
\left(y-y_0\right)=\left(\frac{d y}{d x}\right)_{\left(x 0, y_0\right)} \cdot\left(x-x_0\right)
$ where $\left(\frac{d y}{d x}\right)_{\left(x 0, y_0\right)}$ is the slope.

The equation of normal of a curve is $\left(y-y_0\right) f^{\prime}\left(x_0\right)+\left(x-x_0\right)=0$.

Monotonicity (Increasing and decreasing functions)

A function $f(x)$ is increasing in $[a, b]$ if $f\left(x_2\right) \geq f\left(x_1\right)$ for all $x_2>x_1$, where $x_1, x_2 \in[a, b]$. If a function is differentiable, then $\frac{d}{d x}(f(x)) \geq 0 \quad \forall x \in(a, b)$ is a increasing function while $\frac{d}{d x}(f(x)) > 0 \quad \forall x \in(a, b)$ is a strictly increasing function.

A function $f(x)$ is decreasing in the interval $[a, b]$ if $f\left(x_2\right) \leq f\left(x_1\right)$ for all $x_2>x_1$, where $x_1, x_2 \in[a, b]$. If a function is differentiable, then $\frac{d}{d x}(f(x)) \leq 0 \quad \forall x \in(a, b)$ is a decreasing function while $\frac{d}{d x}(f(x)) < 0 \quad \forall x \in(a, b)$ is a strictly decreasing function.

Maxima and Minima

If $f^{\prime}(x)$ changes sign from positive to negative as $x$ increases through $c$, i.e., if $f^{\prime}(x)>0$ at every point sufficiently close to and to the left of $c$, and $f^{\prime}(x)<0$ at every point sufficiently close to and to the right of $c$, then $c$ is a point of local maxima.

If $f^{\prime}(x)$ changes sign from negative to positive as $x$ increases through $c$, i.e., if $f^{\prime}(x)<0$ at every point sufficiently close to and to the left of $c$, and $f^{\prime}(x)>0$ at every point sufficiently close to and to the right of $c$, then $c$ is a point of local minima.

If at $x=a$ the shape of the curve changes from increasing to decreasing or from decreasing to increasing. Then $\mathrm{x}=\mathrm{a}$ is known as the point of inflection and $f^{\prime \prime}(x)=0$ at $x=a$.

Important Formulae for Application of derivatives

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