Calculus
Limits and Derivatives
Calculus in Math AA begins with the fundamental concepts of limits and derivatives. These form the backbone of differential calculus and are crucial for understanding how functions behave.
Limits
A limit describes the value that a function approaches as the input (usually x) gets closer to a specific value. Formally, we write this as:
$$\lim_{x \to a} f(x) = L$$
This means that as x gets arbitrarily close to a, f(x) gets arbitrarily close to L.
Example
Consider the function $f(x) = \frac{x^2 - 1}{x - 1}$. As x approaches 1, this function approaches 2:
$$\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2$$
Even though the function is undefined at x = 1, the limit exists.
Derivatives
The derivative of a function represents its rate of change. It's defined as the limit of the difference quotient:
$$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$
Geometrically, the derivative at a point gives the slope of the tangent line to the function's graph at that point.
Note
The notation $f'(x)$ is read as "f prime of x" and represents the derivative of f with respect to x.
Increasing/Decreasing Functions and Graphical Interpretation
Understanding how derivatives relate to the behavior of functions is crucial in calculus.
- If $f'(x) > 0$ on an interval, f(x) is increasing on that interval.
- If $f'(x)
< 0$ on an interval, f(x) is decreasing on that interval.
- If $f'(x) = 0$ at a point, that point could be a local maximum, minimum, or inflection point.
Example
For $f(x) = x^3 - 3x^2 + 2$, $f'(x) = 3x^2 - 6x$.
- $f'(x)$ is negative on (0, 2), so f(x) is decreasing there.
- $f'(x)$ is positive on (-∞, 0) and (2, ∞), so f(x) is increasing in these intervals.
Derivatives of Polynomial Functions
Polynomials are the simplest functions to differentiate. The power rule states:
For $f(x) = x^n$, $f'(x) = nx^{n-1}$
This extends to more complex polynomials through the sum rule: the derivative of a sum is the sum of the derivatives.
Example
If $f(x) = 3x^4 - 2x^2 + 5x - 7$, then $f'(x) = 12x^3 - 4x + 5$
Tangents and Normals
The derivative allows us to find equations of tangent and normal lines to curves.
- Tangent line equation at point (a, f(a)): $y - f(a) = f'(a)(x - a)$
- Normal line equation at point (a, f(a)): $y - f(a) = -\frac{1}{f'(a)}(x - a)$
Example
For $f(x) = x^2$ at x = 2: $f(2) = 4$, $f'(2) = 4$ Tangent line: $y - 4 = 4(x - 2)$ Normal line: $y - 4 = -\frac{1}{4}(x - 2)$
Introduction to Integration
Integration is the reverse process of differentiation. The indefinite integral of a function f(x) is written as:
$$\int f(x) dx = F(x) + C$$
Where F(x) is an antiderivative of f(x) and C is an arbitrary constant.
Note
The integral sign $\int$ is an elongated S, representing "sum" as integration can be thought of as the limit of a sum.
Derivatives of Trigonometric, Exponential, and Logarithmic Functions
These special functions have unique derivative rules:
- $\frac{d}{dx}(\sin x) = \cos x$
- $\frac{d}{dx}(\cos x) = -\sin x$
- $\frac{d}{dx}(e^x) = e^x$
- $\frac{d}{dx}(\ln x) = \frac{1}{x}$
Example
If $f(x) = 3\sin x + 2e^x$, then $f'(x) = 3\cos x + 2e^x$
Chain Rule, Product Rule, Quotient Rule
These rules allow us to differentiate more complex functions:
- Chain Rule: $(f(g(x)))' = f'(g(x)) \cdot g'(x)$
- Product Rule: $(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)$
- Quotient Rule: $(\frac{f(x)}{g(x)})' = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$
Example
Using the chain rule: If $y = \sin(x^2)$, then $y' = \cos(x^2) \cdot 2x$
Second Derivatives and Graphical Behavior
The second derivative, $f''(x)$, is the derivative of the derivative. It provides information about the concavity of a function:
- If $f''(x) > 0$, the function is concave up.
- If $f''(x)
< 0$, the function is concave down.
Example
For $f(x) = x^3$, $f'(x) = 3x^2$, and $f''(x) = 6x$. $f''(x)$ is positive when x > 0 and negative when x
< 0, indicating a change in concavity at x = 0.
Optimization and Points of Inflection
Optimization problems involve finding the maximum or minimum values of a function. This often involves finding where $f'(x) = 0$ or where f'(x) is undefined.
Points of inflection occur where the concavity of a function changes. These are points where $f''(x) = 0$ or $f''(x)$ is undefined.
Kinematics Problems
Kinematics applies calculus to motion. Key relationships:
- Position function: s(t)
- Velocity function: v(t) = s'(t)
- Acceleration function: a(t) = v'(t) = s''(t)
Indefinite Integrals of Basic Functions
Indefinite integration is the reverse of differentiation. Some basic integrals:
- $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$)
- $\int \sin x dx = -\cos x + C$
- $\int e^x dx = e^x + C$
Definite Integrals and Area Under Curves
The definite integral represents the area under a curve between two points:
$$\int_a^b f(x) dx = F(b) - F(a)$$
Where F(x) is an antiderivative of f(x).
Continuity, Differentiability, and Limits (AHL)
A function is continuous at a point if the limit of the function as we approach the point from both sides equals the function's value at that point. Differentiability is a stronger condition than continuity; a function is differentiable at a point if its derivative exists at that point.
L'Hôpital's Rule (AHL)
L'Hôpital's Rule is used to evaluate limits that result in indeterminate forms like 0/0 or ∞/∞. It states that:
$$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$$
Implicit Differentiation and Related Rates (AHL)
Implicit differentiation allows us to find derivatives of functions that are not explicitly defined in terms of y = f(x). We differentiate both sides of the equation with respect to x, treating y as a function of x.
Related rates problems involve finding how the rate of change of one quantity relates to the rate of change of another quantity.
Derivatives and Integrals of Inverse Trigonometric Functions (AHL)
Inverse trigonometric functions have specific derivative and integral rules:
- $\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1-x^2}}$
- $\frac{d}{dx}(\arctan x) = \frac{1}{1+x^2}$
Integration Techniques (AHL)
Integration by Substitution
This technique involves making a substitution to simplify the integrand. If $u = g(x)$, then:
$$\int f(g(x))g'(x) dx = \int f(u) du$$
Integration by Parts
This technique is based on the product rule and is useful for integrating products of functions. The formula is:
$$\int u dv = uv - \int v du$$
Volumes of Revolution (AHL)
Volumes of solids formed by rotating a region bounded by a curve around an axis can be calculated using integrals:
- Rotating around x-axis: $V = \pi \int_a^b [f(x)]^2 dx$
- Rotating around y-axis: $V = 2\pi \int_a^b xf(x) dx$
Differential Equations (AHL)
Differential equations involve relationships between a function and its derivatives. In Math AA HL, students study:
Numerical Solutions
These involve approximating solutions using methods like Euler's method.
Separable Equations
These are equations where variables can be separated:
$$\frac{dy}{dx} = f(x)g(y)$$
Homogeneous Equations
These are equations of the form:
$$\frac{dy}{dx} = f(\frac{y}{x})$$
Integrating Factor
This method is used for first-order linear differential equations:
$$\frac{dy}{dx} + P(x)y = Q(x)$$
The integrating factor is $e^{\int P(x) dx}$.
Maclaurin Series (AHL)
Maclaurin series are Taylor series centered at x = 0. They represent functions as infinite sums of terms:
$$f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + ...$$
This comprehensive overview covers the main topics in Calculus for Math AA, providing a solid foundation for both SL and HL students. Remember that practice is key to mastering these concepts!
