Problem 32 - Calculus - AP Edition

Question

The given limit represents \(f^{\prime}(a)\) for some function \(f\) and some number \(a\). Find \(f(x)\) and \(a\) in each case. (a) \(\lim _{h \rightarrow 0} \frac{\cos (\pi+h)+1}{h}\) (b) \(\lim _{x \rightarrow 1} \frac{x^{7}-1}{x-1}\)

Step-by-Step Solution

Answer

(a) \(f(x) = \cos(x), a = \pi\); (b) \(f(x) = x^7, a = 1\).

Step 1: Understanding the Derivative Formulation

Recognize that the limit expressions given are derived from the definition of the derivative. For a function \( f(x) \) at a point \( a \), the derivative is defined as \[ f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \] Compare these with the given limits to identify \( f(x) \) and \( a \).

Step 2: Identify Function and Point for Part (a)

Compare \[ \lim _{h \rightarrow 0} \frac{\cos (\pi+h)+1}{h} \] with the derivative definition.The expression can be written as \[ \lim _{h \rightarrow 0} \frac{\cos (\pi + h) - (-1)}{h} \] This aligns with \( f(a + h) = \cos(\pi + h) \) and \( f(a) = -1 \). This gives \( f(x) = \cos(x) \) and \( a = \pi \).

Step 3: Identify Function and Point for Part (b)

Compare \[ \lim _{x \rightarrow 1} \frac{x^{7}-1}{x-1} \] with the derivative definition. This looks like the limit definition shifted from \( h \) to \( x \). The form \( \frac{f(x) - f(a)}{x-a} \) applies here with \( x \to a \).Let \( f(x) = x^7 \) and \( a = 1 \), so \( f(a) = f(1) = 1 \). This validates the condition, identifying \( f(x) \) and \( a \).

Key Concepts

Derivatives

In calculus, the concept of a derivative is central. It's an essential tool that helps us understand how functions change. Think of it as measuring how fast something moves or changes at a specific point. The derivative of a function at a point gives you the slope of the tangent line to the function's graph at that point.
When we say "finding the derivative," we usually mean applying certain rules or formulas to get a new function that tells us the rate of change at any given point. For example, in the expression \( f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \), we calculate the slope of the secant line as \( h \) approaches zero. This becomes the slope of the tangent line or the derivative at point \( a \).
It's important to recognize this structure when you spot limits like \( \lim_{h \to 0} \frac{\cos (\pi+h)+1}{h} \) or \( \lim_{x \to 1} \frac{x^{7}-1}{x-1} \). They represent the derivative definition applied to different functions.

Trigonometric functions

Trigonometric functions such as sine, cosine, and tangent are fundamental in calculus due to their periodic properties and wide range of applications. These functions help describe phenomena involving waves, oscillations, and cycles.
When we deal with trigonometric functions in the context of derivatives, it's crucial to remember some key derivative rules:

• The derivative of \( \sin(x) \) is \( \cos(x) \).
• The derivative of \( \cos(x) \) is \( -\sin(x) \).

These rules stem from the definition of the derivatives and greatly simplify the process of finding rates of change for trigonometric functions.
For the problem \( \lim _{h \rightarrow 0} \frac{\cos (\pi+h)+1}{h} \), you start by writing it in the form of a derivative. Recognizing that \( f(x) = \cos(x) \) and \( a = \pi \) allows us to see this limit as the derivative of the cosine function.

Power Rule

The power rule is a foundational tool in calculus used to find the derivative of polynomial expressions. It states that if you have a function \( f(x) = x^n \), the derivative \( f'(x) \) is \( nx^{n-1} \). It's simple and extremely powerful for dealing with polynomials!
For example, let's look at the expression \( \lim _{x \rightarrow 1} \frac{x^{7}-1}{x-1} \). This limit resembles the derivative using the power rule. Here, \( f(x) = x^7 \) and \( a = 1 \), which gives us the derivative \( 7x^{6} \) at \( x = 1 \).
Applying the power rule makes it easy to find derivatives quickly without going through the long process of applying the definition of a derivative. It's especially useful when dealing with higher-degree polynomials.