Problem 5 - Calculus for Biology and Medicine

Question

Find the derivative at the indicated point from the graph of \(y=f(x)\). \(f(x)=x^{2} ; x=0\)

Step-by-Step Solution

Answer

The derivative of \( f(x) = x^2 \) at \( x = 0 \) is 0.

Step 1: Identify the Function and Point

We are given the function \( f(x) = x^2 \) and asked to find the derivative at \( x = 0 \). First, recognize that \( f(x) \) is a simple quadratic function.

Step 2: Compute the Derivative of the Function

The derivative of the function \( f(x) = x^2 \) with respect to \( x \) is calculated using the power rule. The power rule states that \( \frac{d}{dx}[x^n] = nx^{n-1} \). For \( f(x) = x^2 \), we find \( f'(x) = 2x^{2-1} = 2x \).

Step 3: Evaluate the Derivative at the Given Point

Next, substitute \( x = 0 \) into the derivative \( f'(x) = 2x \). This gives us \( f'(0) = 2(0) = 0 \).

Step 4: Interpret the Result

The derivative \( f'(0) = 0 \) indicates that at \( x = 0 \), the slope of the tangent to the graph of \( f(x) = x^2 \) is zero. This means the graph is flat at this point, confirming that \( x = 0 \) is the vertex of the parabola described by \( y = x^2 \).

Key Concepts

Power Rule

The power rule is a fundamental concept used to find the derivative of functions of the form \(f(x) = x^n\). It simplifies the differentiation process, especially for polynomial functions.

• The general form of the power rule is written as \(\frac{d}{dx}[x^n] = nx^{n-1}\).
• This rule states that to find the derivative, you multiply the exponent \(n\) by the base \(x\) raised to the power of \(n-1\).

Let's apply it to our original function \(f(x) = x^2\). Using the power rule, the derivative \(f'(x)\) becomes \(2x^{2-1} = 2x\).
This rule makes it quick and easy to calculate derivatives of terms like \(x^3\), \(x^4\), or any power \(n\). Whenever you see a power of \(x\), the power rule is your go-to tool for finding how the function's slope changes.

Quadratic Function

A quadratic function is a type of polynomial function where the highest power of the variable is 2. This forms the general function \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\).
For the given example, \(f(x) = x^2\), we have a simple quadratic function with \(a = 1\), \(b = 0\), and \(c = 0\). The graph of any quadratic function is a parabola, which can open upwards or downwards depending on the sign of \(a\).

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), the parabola opens downwards.

For \(f(x) = x^2\), the parabola opens upwards, and its vertex, the highest or lowest point, is at the origin \((0, 0)\). Quadratics like \(x^2\) are essential in modeling various real-world scenarios in physics and engineering, thanks to their simplicity and predictable curvature.

Tangent

In calculus, a tangent refers to a straight line that just "touches" a curve at a specific point without crossing it. At that point, the tangent line's slope equals the derivative of the function at that specific point.For our function \(f(x) = x^2\), the tangent line at \(x = 0\) has a particular significance. When we computed the derivative and found \(f'(0) = 0\), it told us something crucial about the shape of the parabola at that point.

• A slope of zero for the tangent line means the line is perfectly horizontal at \(x=0\).
• This implies that the graph of \(f(x) = x^2\) is flat at the origin.

This horizontal tangent signifies the vertex of the parabola, marking it as a critical point where the curve transitions from decreasing to increasing. Understanding tangents helps us analyze functions' behaviors and provides insights into their maxima, minima, and points of inflection.