Problem 36 - Calculus for Biology and Medicine

Question

The following limit represents the derivative of a function \(f\) at the point \((a, f(a))\) : $$ \lim _{h \rightarrow 0} \frac{4(a+h)^{3}-4 a^{3}}{h} $$

Step-by-Step Solution

Answer

The derivative \( f'(a) = 12a^2 \).

Step 1: Identify the Function

The limit given in the problem represents the derivative of the function at a certain point. Given \( \lim _{h \rightarrow 0} \frac{4(a+h)^{3}-4 a^{3}}{h} \), it is structured as a derivative definition. The function, therefore, is \( f(x) = 4x^3 \).

Step 2: Set Up the Derivative Formula

Recall the definition of the derivative: \( f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \). Substitute \( f(a) = 4a^3 \) and \( f(a+h) = 4(a+h)^3 \) into the formula. The limit becomes \( \lim _{h \rightarrow 0} \frac{4(a+h)^{3}-4 a^{3}}{h} \), which matches the given limit.

Step 3: Expand the Expression

Expand the expression \( 4(a+h)^3 \). This requires using the binomial expansion: \( (a+h)^3 = a^3 + 3a^2h + 3ah^2 + h^3 \). Substituting back into the expression: \[ 4(a+h)^3 = 4(a^3 + 3a^2h + 3ah^2 + h^3) \] which becomes \[ 4a^3 + 12a^2h + 12ah^2 + 4h^3 \].

Step 4: Simplify the Limit Expression

Replace the expanded form back into the limit expression:\[ \lim_{h \to 0} \frac{4a^3 + 12a^2h + 12ah^2 + 4h^3 - 4a^3}{h} \]Simplify by canceling out \( 4a^3 \) terms:\[ \lim_{h \to 0} \frac{12a^2h + 12ah^2 + 4h^3}{h} \].

Step 5: Factor out an h and Simplify

Factor an \( h \) from the numerator:\[ \lim_{h \to 0} \frac{h(12a^2 + 12ah + 4h^2)}{h} \]Cancel the \( h \) in the numerator and denominator:\[ \lim_{h \to 0} (12a^2 + 12ah + 4h^2) \].

Step 6: Evaluate the Limit

Evaluate the limit as \( h \to 0 \):\[ 12a^2 + 12a(0) + 4(0)^2 = 12a^2 \].Thus, the derivative \( f'(a) = 12a^2 \).

Key Concepts

Limit

In calculus, the concept of a limit is foundational when discussing derivatives. A limit helps us understand the behavior of a function as its input approaches some value. When we say \( h \to 0 \), we are examining what happens to the function expression as \( h \) becomes infinitesimally small. This is crucial in calculating the derivative, which represents the slope of the tangent line to the curve of the function at a specific point. By analyzing the limit form \( \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \), we determine the rate of change of the function \( f \) at the point \( a \). Therefore, limits form the backbone of understanding how a derivative captures the instantaneous rate of change.

Binomial Expansion

The binomial expansion is a method used to expand expressions that are raised to a power, such as \((a+h)^3\). It is a powerful algebraic tool that simplifies polynomials, especially useful when applying the definition of derivatives.The binomial theorem states that \((x+y)^n\) can be expanded into a sum of terms of the form \( \binom{n}{k}x^{n-k}y^k \). For the specific case of \((a+h)^3\), it expands to:

• \(a^3\)
• \(3a^2h\)
• \(3ah^2\)
• \(h^3\)

Substituting back into a derivative problem, it allows for a straightforward simplification crucial for eliminating terms and isolating the impact of \(h\) as it approaches zero.

Derivative Definition

The definition of the derivative is a limit which can be written as \( f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \). This mathematical expression captures the concept of an instantaneous rate of change at the specific point \( a \) on the function \( f(x) \).It involves:

• Substituting \( f(a) \) for the original function value at point \( a \).
• Substituting \( f(a+h) \) for the function value at a slight increment \( h \) past \( a \).

By calculating this limit, we obtain the derivative, which is essentially the slope of the tangent to the function at that point.

Differentiation Techniques

Differentiation is the process of finding a derivative, and there are various techniques for conducting differentiation depending on the form of the function.Here are a few key techniques:

• Power Rule: Used for functions of the form \(x^n\), where the derivative is \(nx^{n-1}\).
• Product Rule: Useful when differentiating products of functions, given by \((fg)' = f'g + fg'\).
• Quotient Rule: Employed for quotients of functions, described as \((\frac{f}{g})' = \frac{f'g - fg'}{g^2}\).
• Chain Rule: Used for composite functions \((f(g(x)))\), where the derivative is \(f'(g(x))g'(x)\).

Employing these techniques helps streamline finding derivatives, allowing for efficient simplification of complex expressions especially when combined with calculating limits.