Find the Difference Quotient f(x)=3x^3-7
The given problem asks for the computation of the difference quotient of the function f(x) = 3x^3 - 7. The difference quotient is used in calculus to give an average rate of change of the function between two points. It is usually expressed in the form:
(f(x + h) - f(x)) / h
Here, "h" represents an increment in the x-value, and the quotient measures how much the function value changes for a small change in x around the point x. The difference quotient is a fundamental concept in the definition of the derivative of a function, which at the limit as h approaches zero, gives the instantaneous rate of change of the function at x.
$f \left(\right. x \left.\right) = 3 x^{3} - 7$
Answer
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: $f(x + h) = 3(x^3 + 3x^2h + 3xh^2 + h^3) - 7$.
Step:2.1.2.1.2
Distribute the coefficient $3$ across the terms: $f(x + h) = 3x^3 + 9x^2h + 9xh^2 + 3h^3 - 7$.
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: $f(x + h) = 3x^3 + 9x^2h + 9xh^2 + 3h^3 - 7$.
Step:2.1.2.1.4
Eliminate any unnecessary parentheses.
Step:2.1.2.2
The simplified function at $x + h$ is $3x^3 + 9x^2h + 9xh^2 + 3h^3 - 7$.
Step:2.2
Rearrange terms if necessary.
Step:2.3
Determine $f(x)$ and $f(x + h)$: $f(x) = 3x^3 - 7$ and $f(x + h) = 3h^3 + 9h^2x + 9hx^2 + 3x^3 - 7$.
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 $3h^3 + 9h^2x + 9hx^2$.
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.
