Question

A function \(f\) and \(a\) value \(a\) are given. Approximate the limit of the difference quotient, \(\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h},\) using \(h=\pm 0.1, \pm 0.01 .\) $$ f(x)=9 x+0.06, \quad a=-1 $$

Short answer

The approximate limit is 9.

Step-by-step solution

Identify the Function and Values

We are given the function \( f(x) = 9x + 0.06 \) and the value of \( a = -1 \). We need to approximate the limit of the difference quotient \( \lim_{h \rightarrow 0} \frac{f(a+h) - f(a)}{h} \) using \( h = \pm 0.1, \pm 0.01 \).

Calculate \( f(a) \)

Substitute \( a = -1 \) into the function to find \( f(a) \):\[f(-1) = 9(-1) + 0.06 = -9 + 0.06 = -8.94\]

Compute f(a+h) for Different Values of h

We will compute \( f(a+h) \) for each \( h \) value:1. \( h = 0.1: f(-1 + 0.1) = f(-0.9) = 9(-0.9) + 0.06 = -8.1 + 0.06 = -8.04 \)2. \( h = -0.1: f(-1 - 0.1) = f(-1.1) = 9(-1.1) + 0.06 = -9.9 + 0.06 = -9.84 \)3. \( h = 0.01: f(-1 + 0.01) = f(-0.99) = 9(-0.99) + 0.06 = -8.91 + 0.06 = -8.85 \)4. \( h = -0.01: f(-1 - 0.01) = f(-1.01) = 9(-1.01) + 0.06 = -9.09 + 0.06 = -9.03 \)

Compute the Difference Quotient for Each h

Compute the difference quotient \( \frac{f(a+h) - f(a)}{h} \) for each \( h \):1. For \( h = 0.1:\) \[ \frac{-8.04 - (-8.94)}{0.1} = \frac{0.9}{0.1} = 9 \]2. For \( h = -0.1:\) \[ \frac{-9.84 - (-8.94)}{-0.1} = \frac{-0.9}{-0.1} = 9 \]3. For \( h = 0.01:\) \[ \frac{-8.85 - (-8.94)}{0.01} = \frac{0.09}{0.01} = 9 \]4. For \( h = -0.01:\) \[ \frac{-9.03 - (-8.94)}{-0.01} = \frac{-0.09}{-0.01} = 9 \]

Interpret the Results

For each value of \( h \), the difference quotient is 9. This consistency supports that the limit \( \lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h} \) approaches 9. Therefore, the derivative of \( f(x) \) at \( a = -1 \) is approximately 9.

Key concepts

Limit of a Function

In calculus, the concept of the limit is fundamental. It's like a bridge that connects discrete values with continuous behaviors. A limit of a function at a particular point describes how the function behaves as it gets closer to that point. When we say the limit of a function as we approach a value, we mean what value the function is getting closer to.

Think of it as moving towards a specific spot on a number line. Even if you never actually "touch" that point, understanding your path is what limits are all about. In this context, the limit of the difference quotient as \(h\) approaches zero helps us determine the rate of change - which leads to the concept of a derivative!

Key aspects of limits include:

• Getting close: It's about values getting closer to a target number without necessarily ever reaching it.
• Understanding behavior: Limits explore how functions behave as inputs approach specific values.
• Crucial for calculus: Limits underpin much of calculus, bringing the idea of continuity into focus.

Derivative

The derivative is a cornerstone concept in calculus that measures how a function changes as its input changes. Imagine driving a car. The speed you're traveling is a derivative of your position, exploring how fast your position changes over time.

Mathematically, the derivative at any point is calculated using the difference quotient we've discussed. When the limit of this difference quotient as \(h\) approaches zero exists, the derivative is simply that limit. This operation gives us the slope of the tangent line to the curve of a function at any point. Such an interpretation is pivotal for understanding rates of change.
Some important points about derivatives include:

• Rate of change: Derivatives measure how quickly or slowly a quantity changes.
• Slope interpretation: Derivatives provide the slope of the function's graph at any point.
• Ubiquitous application: They're used in physics, engineering, and economics to model real-world scenarios.

Approximation Methods

Approximation methods are about finding close estimations of mathematical values or behaviors. Instead of calculating an exact answer, these methods give us a very close 'near-enough' result. In the context of our exercise, by taking small values of \(h\), we can approximate the derivative of a function.

The method we employed involved calculating the difference quotient for small values of \(h\) (like \(+0.1\) and \(-0.01\)). This lets us observe consistent values that suggest what happens as \(h\) approaches zero. This kind of estimation is essential because in many real-life situations, exact solutions are either costly or impossible to find.
Features of approximation methods:

• Close estimates: They provide values close enough to the exact answer to be useful.
• Feasibility: They make it possible to make calculations when exact methods are complicated.
• Iterative approach: Often, these methods improve in accuracy with more data or iterations.