Sure, let's consider a classic example of a function in calculus: the quadratic function.

Example: Quadratic Function

A quadratic function is a polynomial function of degree 2, and it can be written in the form:

$f(x) = ax^2 + bx + c$

where $a$, $b$, and $c$ are constants, and $a \neq 0$.

Let's work with a specific example:

$f(x) = 2x^2 - 3x + 1$

Key Calculus Concepts Applied to this Function

1. Finding the Derivative (First Derivative):

   • The first derivative of a function gives us the slope of the tangent line at any point $x$.
   • For $f(x) = 2x^2 - 3x + 1$: $f'(x) = \frac{d}{dx}(2x^2) - \frac{d}{dx}(3x) + \frac{d}{dx}(1)$ $f'(x) = 4x - 3$

2. Critical Points:

   • Critical points occur where the first derivative is zero or undefined.
   • Set $f'(x) = 0$: $4x - 3 = 0$ $x = \frac{3}{4}$

3. Second Derivative:

   • The second derivative provides information about the concavity of the function.

   • For $f(x) = 2x^2 - 3x + 1$: $f''(x) = \frac{d}{dx}(4x - 3)$ $f''(x) = 4$

   • Since $f''(x) = 4 > 0$, the function is concave up, and the critical point at $x = \frac{3}{4}$ is a local minimum.

4. Vertex of the Parabola:

   • The vertex form of a quadratic function is another useful representation.

   • The vertex can be found using the formula $x = -\frac{b}{2a}$: $x = -\frac{-3}{2 \cdot 2}$ $x = \frac{3}{4}$

   • Substitute $x = \frac{3}{4}$ back into $f(x)$ to find the y-coordinate of the vertex: $f\left(\frac{3}{4}\right) = 2\left(\frac{3}{4}\right)^2 - 3\left(\frac{3}{4}\right) + 1$ $f\left(\frac{3}{4}\right) = 2 \cdot \frac{9}{16} - 3 \cdot \frac{3}{4} + 1$ $f\left(\frac{3}{4}\right) = \frac{18}{16} - \frac{9}{4} + 1$ $f\left(\frac{3}{4}\right) = \frac{18}{16} - \frac{36}{16} + \frac{16}{16}$ $f\left(\frac{3}{4}\right) = \frac{18 - 36 + 16}{16}$ $f\left(\frac{3}{4}\right) = \frac{-2}{16}$ $f\left(\frac{3}{4}\right) = -\frac{1}{8}$

   • So, the vertex of the function $f(x) = 2x^2 - 3x + 1$ is at $\left(\frac{3}{4}, -\frac{1}{8}\right)$.

Summary:

The function $f(x) = 2x^2 - 3x + 1$ has:

• A derivative $f'(x) = 4x - 3$
• A critical point at $x = \frac{3}{4}$, which is a local minimum
• A vertex at $\left(\frac{3}{4}, -\frac{1}{8}\right)$

Would you like more details or have any questions? Here are some related questions to further explore:

1. How do you find the derivative of a polynomial function?
2. What is the significance of the second derivative in calculus?
3. Can you explain the process of completing the square for a quadratic function?
4. How do you determine if a critical point is a maximum or minimum?
5. What are inflection points and how do you find them?
6. How do you graph a quadratic function using its vertex and intercepts?
7. Can you provide an example of a real-world application of quadratic functions?
8. How do you solve a quadratic equation using the quadratic formula?

Tip: Understanding the graphical representation of a function alongside its algebraic form can significantly enhance your comprehension of calculus concepts.