\[ f(x) = 4+ 3x -x^{2}, \space \dfrac{f(3+h) – f(3)}{h} \]

This question belongs to the **calculus** domain, and the aim is to **understand** the difference **quotient** and the practical **application** where it is being used.

The **difference quotient** is the term for the expression:

\[ \dfrac{f(x+h)-f(h)}{h}\]

Where, when the **limit** h approaches $\rightarrow$ 0, delivers the **derivative** of the **function** $f$. As the expression itself **explains** that it is the **quotient** of the difference of the values of the **function** by the difference of the **affiliated** values of its **argument.** The rate of **change** of the function throughout **length** $h$ is called as the **difference quotient.** The limit of the difference quotient is the **instantaneous** rate of change.

In **numerical differentiation** the difference quotients are used as **approximations,** In time **discretization,** the difference quotient may also find **relevance.** Where the **width** of the time step is input as the **value** $h$.

## Expert Answer

Given the **function** $f(x)$ is:

\[ f(x) = 4+3x-x^{2}\]

The difference **quotient** is given as:

\[ \dfrac{f(3+h) – f(3)}{h} \]:

First, we will compute the **expression** for $f(3+h)$:

\[ f(x) = 4+3x-x^{2}\]

\[ f(3+h) = 4+ 3(3+h)- (3+h)^{2} \]

Expanding $(3+h)^{2}$ using the **formula** $(a+b)^2 = a^2 + b^2 + 2ab$

\[ f(3+h) = 4+ 9+3h- (3^2 + h^2 + 2(3)(h) \]

\[ f(3+h) = 4+ 9+3h- (3^2 + h^2 + 2(3)(h)) \]

\[ f(3+h) = 13+3h – (9+ h^2 + 6(h)) \]

\[ f(3+h) = 13+3h -9 -h^2 -6(h)) \]

\[ f(3+h) = 4 -3h -h^2 \]

Now **computing** the expression for $f(3)$:

\[ f(x) = 4+3x- x^{2}\]

\[ f(3) = 4+3(3)- (3)^{2}\]

\[ f(3) = 4+9- 9\]

\[ f(3) = 4\]

Now **insert** the expressions in the **difference** quotient:

\[= \dfrac{f(3+h) – f(3)} {h} \]

\[ =\dfrac{(4 -3h -h^2) – 4} {h} \]

\[ =\dfrac{4 -3h -h^2 -4} {h} \]

\[ = \dfrac{h(-3 -h)} {h}\]

\[ = -3 -h \]

## Numerical Answer

The **difference quotient** $\dfrac{f(3+h) – f(3)}{h}$ for the function $ f(x) = 4+3x-x^{2}$ is $-3 -h$.

## Example

Given the **function:**

\[ f(x) = -x^3, \space \dfrac{f(a+h) – f(a)}{h}\]

find the exact difference **quotient** and simplify your answer.

Given the function $f(x)$ is:

\[ f(x) = -x^ {3} \]

The **difference** quotient is given as:

\[ \dfrac{f(a+h) – f(a)} {h} \]

Firstly we will compute the **expression** for $f(a+h)$:

\[ f(x) = -x^{3} \]

\[ f(a+h) = – (a+h)^ {3} \]

Expanding $(3+h)^{2}$ using the **formula** $(a+b)^3 = a^3 + b^3 + 3a^2b + 3ab^2$

\[ f(a+h) = – (a^3 + h^3 + 3a^2h + 3ah^2) \]

Now computing the **expression** for $f(a)$:

\[ f(x) = – x^{3}\]

\[ f(a) = -a^{3}\]

Now insert the expressions in the **difference** quotient:

\[= \dfrac{f(a+h) – f(a)}{h} \]

\[ =\dfrac{- (a^3 + h^3 + 3a^2h + 3ah^2) – (-a^{3})} {h} \]

\[ =\dfrac{ -a^3 -h^3 -3a^2h -3ah^2 +a^{3}} {h} \]

\[ =\dfrac{ -h^3 -3a^2h -3ah^2 } {h} \]

\[ =\dfrac{h( -h^2 -3a^2 -3ah) } {h} \]

\[ = -3a^2 -3ah -h^2 \]

The **difference quotient** $\dfrac{f(a+h) – f(a)}{h}$ for the function $ f(x) = -x^{3}$ is $ -3a^2 -3ah -h^2 $.