Find the Difference Quotient f(x)=3x^3-7

Solution:

Step:1

Start with the difference quotient formula: $\frac{f(x+h)-f(x)}{h}$.

Step:2

Determine the function values needed.

Step:2.1

Calculate $f(x+h)$.

Step:2.1.1

Substitute $x$ with $x+h$ in the function: $f(x+h)=3(x+h{)}^3-7$.

Step:2.1.2

Expand and simplify the expression.

Step:2.1.2.1

Expand each term individually.

Step:2.1.2.1.1

Apply the Binomial Theorem to expand: $f(x+h)=3(x^3+3x^{2}h+3x{h}^2+h^3)-7$.

Step:2.1.2.1.2

Distribute the coefficient $3$ across the terms: $f(x+h)=3x^3+9x^{2}h+9x{h}^2+3h^3-7$.

Step:2.1.2.1.3

Complete the simplification process.

Step:2.1.2.1.3.1

Combine like terms where applicable.

Step:2.1.2.1.3.2

The expression is now fully simplified: $f(x+h)=3x^3+9x^{2}h+9x{h}^2+3h^3-7$.

Step:2.1.2.1.4

Eliminate any unnecessary parentheses.

Step:2.1.2.2

The simplified function at $x+h$ is $3x^3+9x^{2}h+9x{h}^2+3h^3-7$.

Step:2.2

Rearrange terms if necessary.

Step:2.3

Determine $f(x)$ and $f(x+h)$: $f(x)=3x^3-7$ and $f(x+h)=3h^3+9h^{2}x+9h{x}^2+3x^3-7$.

Step:3

Insert the function values into the difference quotient.

Step:4

Simplify the expression.

Step:4.1

Work on the numerator.

Step:4.1.1

Apply the distributive property to expand and combine like terms.

Step:4.1.2

Simplify by combining and canceling out terms.

Step:4.1.3

The simplified numerator becomes $3h^3+9h^{2}x+9h{x}^2$.

Step:4.1.4

Factor out the common factor of $h$ in the numerator.

Step:4.1.5

The factored form is $3h(h^2+3hx+3x^2)$.

Step:4.2

Reduce the fraction by canceling common factors.

Step:4.3

Finish simplifying to get the final result: $3h^2+9hx+9x^2$.

Step:5

The difference quotient for $f(x)=3x^3-7$ is $\frac{3h^2+9hx+9x^2}{1}$, which simplifies to $3h^2+9hx+9x^2$.

Knowledge Notes:

The difference quotient is a fundamental concept in calculus, representing the average rate of change of a function over a small interval $h$. It is a precursor to the derivative, which is the limit of the difference quotient as $h$ approaches zero.

The Binomial Theorem is a key tool in algebra for expanding expressions of the form $(x+y{)}^n$. It expresses the expansion in terms of the sum of binomial coefficients multiplied by the terms $x$ and $y$ raised to varying powers.

The distributive property is an algebraic property used to simplify expressions by multiplying a single term by each term within a set of parentheses.

Factoring is the process of breaking down an expression into products of simpler expressions. It is often used to simplify expressions and solve equations.

Simplifying expressions involves combining like terms, canceling common factors, and reducing fractions to their simplest form. This process is essential for making complex algebraic expressions more manageable and for finding solutions to equations.