Calculus Study Notes: Quadratic Polynomials, Derivatives, and Applications

Quadratic Polynomials and Their Derivatives

Finding a Quadratic Polynomial Given Function and Derivative Values

Quadratic polynomials are functions of the form f(x) = ax^2 + bx + c. Given values for the function and its derivatives at specific points, we can determine the coefficients a, b, and c.

• General Form: $f(x) = ax^2 + bx + c$

• First Derivative: $f'(x) = 2ax + b$

• Second Derivative: $f''(x) = 2a$

To find the coefficients, set up equations using the given values:

1. Plug in the given x-values into $f(x)$, $f'(x)$, and $f''(x)$ to form a system of equations.

2. Solve for a, b, and c using substitution or elimination.

Example: Find a quadratic polynomial $f(x) = ax^2 + bx + c$ such that $f(0) = 5$, $f'(0) = 3$, and $f''(x) = 6$.

• $f(0) = c = 5$

• $f'(0) = b = 3$

• $f''(x) = 2a = 6 \implies a = 3$

Final Answer: $f(x) = 3x^2 + 3x + 5$

Graphical Representation of Quadratic Functions

Sketching and Interpreting the Graph

Quadratic functions graph as parabolas. The sign of the leading coefficient (a) determines if the parabola opens upwards (a > 0) or downwards (a < 0).

• Vertex: The vertex of $f(x) = ax^2 + bx + c$ is at $x = -\frac{b}{2a}$.

• Axis of Symmetry: The line $x = -\frac{b}{2a}$.

• Y-intercept: At $x = 0$, $f(0) = c$.

Example: For $f(x) = 3x^2 + 3x + 5$, the vertex is at $x = -\frac{3}{6} = -0.5$.

Application: Modeling with Exponential Functions

Antibiotic Concentration in Bloodstream

Exponential functions are often used to model decay processes, such as the concentration of a drug in the bloodstream over time.

• General Form: $C(t) = C_0 e^{-kt}$, where $C_0$ is the initial concentration and $k$ is the decay constant.

• Interpretation: The function decreases over time, modeling how the drug is eliminated from the body.

Example: If the concentration is given by $C(t) = 25e^{-0.3t}$, then:

• Initial concentration: $C(0) = 25$

• Concentration after 6 hours: $C(6) = 25e^{-0.3 \times 6} = 25e^{-1.8}$

Graph: The graph of $C(t)$ is a decreasing exponential curve starting at $C_0$ and approaching zero as $t$ increases.

Summary Table: Quadratic and Exponential Functions

Function Type	General Form	Key Properties	Example
Quadratic	$f(x) = ax^2 + bx + c$	Parabola, vertex at $x = -\frac{b}{2a}$, axis of symmetry, y-intercept at $c$	$f(x) = 3x^2 + 3x + 5$
Exponential Decay	$C(t) = C_0 e^{-kt}$	Decreases over time, models decay/elimination	$C(t) = 25e^{-0.3t}$

Additional info:

• Some steps and explanations were inferred for clarity and completeness, as the original notes were brief and partially fragmented.

• Graphical sketches and some color-coded highlights were interpreted as key points and examples.