91Ó°ÊÓ

The difference quotient is a central concept in calculus, used to find the instantaneous rate of change of a function, or its derivative. The formula for the difference quotient is given as:\[f^{\prime}(x) = \lim _{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}\]Here's how it works:

• You take the function value at \(x\) and at \(x + h\).
• Subtract these function values to see how much the function changes over the interval \([x, x+h]\).
• Divide by \(h\) to find the average rate of change, or slope, of the function over that interval.
• Finally, take the limit as \(h\) approaches zero.This gives you the actual instantaneous rate of change, or the derivative.

Breaking it down step by step with a specific function, like in our given exercise with \(g(x) = x^4 + x^2\), helps see the method clearly. The goal is to simplify the expression until the terms involving \(h\) vanish as \(h\) approaches zero. This result reveals the derivative of the function, a crucial part of calculus.

Binomial expansion is a technique used to expand expressions that are raised to a power. It's based on the binomial theorem, which allows us to express powers of a sum in terms of sums and products of the components. In our case, the binomial expansion is used to simplify expressions like \[(x + h)^4\] and \[(x + h)^2\]. Using the binomial theorem:

• \((x + h)^4 = x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4 \)
• \((x + h)^2 = x^2 + 2xh + h^2\)

This expansion is essential because it allows the calculation of \(g(x + h)\), an important step in finding the derivative using the difference quotient. Notice how each term in the expansion represents a fractional part of the overall change, with varying powers of \(h\) that eventually diminish once you apply the limit.

The limit of a function is a fundamental concept in calculus, used to define the behavior of a function as its input approaches a certain value. Limits are crucial for defining derivatives and integrals.In our exercise, the limit:\[\lim_{h \to 0} h(4x^3 + 6x^2h + 4xh^2 + h^3 + 2x + h) = 4x^3 + 2x\]shows us what happens to the result of the difference quotient as \(h\) becomes incredibly small, approaching zero.Important aspects of limits include:

• The ability to understand the behavior of functions at points that might not be explicitly defined.
• Helps deduce the derivative, highlighting how small changes in input affect the function's output.

In this exercise, notice how the terms involving \(h\) disappear because \(h\) approaches zero. This simplification allows us to isolate the significant components of the derivative, in this case, \(4x^3 + 2x\), evidencing the power of limits in determining precise mathematical expressions.