Problem 17 - Precalculus with Limits

Question

In Exercises 17-22, find a formula for the slope of the graph of \(f\) at the point \((x, f(x))\). Then use it to find the slope at the two given points. \(f(x) = 4 - x^2\) (a) \((0, 4)\) (b) \((-1, 3)\)

Step-by-Step Solution

Answer

The slope of the function \(f(x) = 4 - x^2\) at points (0, 4) and (-1, 3) are 0 and 2 respectively.

Step 1: Calculate Derivative

To find the derivative, \(f'(x)\), of the given function, \(f(x) = 4 - x^2\), apply the power rule, which states that the derivative of \(x^n\) is \(n \cdot x^{n - 1}\). The derivative of the constant 4 is zero. Hence, the derivative formula becomes: \(f'(x) = -2x\)

Step 2: Find slope at (0,4)

Now, use this formula to calculate the derivative of \(f\) at the point (0, 4). Replace \(x\) in \(f'(x) = -2x\) with 0: \(f'(0) = -2 \cdot 0 = 0\)

Step 3: Find slope at (-1,3)

Similarly, to find the slope at the point (-1, 3), substitute \(x\) with -1: \(f'(-1) = -2 \cdot -1 = 2\).

Key Concepts

Slope

The concept of the slope can be thought of as the steepness or incline of a line. In math, when we talk about the slope at a specific point on a curve, we're looking at the rate of change or the derivative of the function at that point.

Imagine you're hiking up a hill. If the slope is very steep, you'll feel it as you climb. A zero slope means you're walking on flat ground. In our exercise, the slope at a given point tells us how steep the tangent line to the function is at that point.

For a function like \(f(x) = 4 - x^2\), finding the slope involves calculating its derivative, which is essentially a formula that gives us the slope at any \(x\) value.

Power Rule

The power rule is a straightforward technique used to calculate the derivative of functions that are powers of \(x\). It’s like a shortcut that makes differentiation quicker and simpler.

The rule states that if you have a term \(x^n\), its derivative is \(n \cdot x^{n-1}\). This means you multiply by the power and then reduce the power by one.

In our exercise with \(f(x) = 4 - x^2\), applying the power rule to \(-x^2\) gives us \(-2x\). You simply bring the 2 in front, multiply by \(x\), and drop the power to 1. The derivative of the constant 4 is zero, because constants don’t change and thus have no slope.

Function

A function is like a machine where you input a value and get an output. It assigns every input value exactly one output value. In our exercise, the function is \(f(x) = 4 - x^2\).

Here's how it works: if you put in a specific \(x\) value, you’ll get a corresponding \(f(x)\), or \(y\) value that tells you a point’s position on the graph.

Depending on the formula, a function can curve, peak, dip, or create all sorts of shapes on a graph. Understanding functions helps uncover patterns, relationships, and predict changes.

• Functions can be linear or non-linear, continuous, or discrete.
• They are essential for modeling real-world situations.

By finding the derivative of the function \(f(x) = 4 - x^2\), we can determine how the graph behaves at different \(x\) values.