AP Calculus Formulas
Limits:
- Limit Definition: \( \lim_{{x \to c}} f(x) \)
- Sum/Difference Rule: \( \lim_{{x \to c}} [f(x) \pm g(x)] = \lim_{{x \to c}} f(x) \pm \lim_{{x \to c}} g(x) \)
- Product Rule: \( \lim_{{x \to c}} [f(x) \cdot g(x)] = \lim_{{x \to c}} f(x) \cdot \lim_{{x \to c}} g(x) \)
- Quotient Rule: \( \lim_{{x \to c}} \frac{{f(x)}}{{g(x)}} = \frac{{\lim_{{x \to c}} f(x)}}{{\lim_{{x \to c}} g(x)}} \) (if \( \lim_{{x \to c}} g(x) \neq 0 \))
Continuity:
- Definition of Continuity: A function \( f(x) \) is continuous at a point \( c \) if \( \lim_{{x \to c}} f(x) \) exists and equals \( f(c) \).
- Types of Discontinuity: Removable, Jump, Infinite, Essential.
- Intermediate Value Theorem: If \( f(x) \) is continuous on \([a, b]\) and \( k \) is between \( f(a) \) and \( f(b) \), then there exists \( c \) in \((a, b)\) such that \( f(c) = k \).
\( f(x) = x^2 + 3x + 2 \)
AP Calculus Differentiation Formulas
Differentiation Formulas:
- Derivative Definition: \( \frac{{d}}{{dx}} [f(x)] = f'(x) \) or \( \frac{{df}}{{dx}} \)
- Constant Rule: \( \frac{{d}}{{dx}} [c] = 0 \) (where \( c \) is a constant)
- Power Rule: \( \frac{{d}}{{dx}} [x^n] = nx^{n-1} \) (where \( n \) is any real number)
- Sum/Difference Rule: \( \frac{{d}}{{dx}} [f(x) \pm g(x)] = \frac{{df}}{{dx}} \pm \frac{{dg}}{{dx}} \)
- Product Rule: \( \frac{{d}}{{dx}} [f(x) \cdot g(x)] = f'(x) \cdot g(x) + f(x) \cdot g'(x) \)
- Quotient Rule: \( \frac{{d}}{{dx}} \left[\frac{{f(x)}}{{g(x)}}\right] = \frac{{f'(x) \cdot g(x) - f(x) \cdot g'(x)}}{{[g(x)]^2}} \) (where \( g(x) \neq 0 \))
- Chain Rule: If \( y = f(u) \) and \( u = g(x) \), then \( \frac{{dy}}{{dx}} = \frac{{dy}}{{du}} \cdot \frac{{du}}{{dx}} \)
- Derivative of Sine: \( \frac{{d}}{{dx}} [\sin(x)] = \cos(x) \)
- Derivative of Cosine: \( \frac{{d}}{{dx}} [\cos(x)] = -\sin(x) \)
- Derivative of Tangent: \( \frac{{d}}{{dx}} [\tan(x)] = \sec^2(x) \)
- Derivative of Exponential: \( \frac{{d}}{{dx}} [e^x] = e^x \)
- Derivative of Natural Logarithm: \( \frac{{d}}{{dx}} [\ln(x)] = \frac{1}{x} \)
- Derivative of Arcsine: \( \frac{{d}}{{dx}} [\arcsin(x)] = \frac{1}{{\sqrt{1 - x^2}}} \) (for \( |x| < 1 \))
- Derivative of Arccosine: \( \frac{{d}}{{dx}} [\arccos(x)] = -\frac{1}{{\sqrt{1 - x^2}}} \) (for \( |x| < 1 \))
- Derivative of Arctangent: \( \frac{{d}}{{dx}} [\arctan(x)] = \frac{1}{{1 + x^2}} \)
AP Calculus Integration Formulas
Integration Formulas:
- Indefinite Integral: \( \int f(x) \, dx \) (Represents the antiderivative of \( f(x) \))
- Constant Multiple Rule: \( \int cf(x) \, dx = c \int f(x) \, dx \) (where \( c \) is a constant)
- Power Rule: \( \int x^n \, dx = \frac{{x^{n+1}}}{{n+1}} + C \) (where \( n \) is not equal to -1)
- Sum/Difference Rule: \( \int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx \)
- Integration by Substitution: \( \int f(g(x)) \cdot g'(x) \, dx = \int f(u) \, du \) (where \( u = g(x) \))
- Integration by Parts: \( \int u \, dv = uv - \int v \, du \)
- Trigonometric Integrals:
- Integral of Sine: \( \int \sin(x) \, dx = -\cos(x) + C \)
- Integral of Cosine: \( \int \cos(x) \, dx = \sin(x) + C \)
- Integral of Secant Squared: \( \int \sec^2(x) \, dx = \tan(x) + C \)
- Integral of Sine: \( \int \sin(x) \, dx = -\cos(x) + C \)
- Exponential and Logarithmic Integrals:
- Integral of Exponential: \( \int e^x \, dx = e^x + C \)
- Integral of Natural Logarithm: \( \int \ln(x) \, dx = x \ln(x) - x + C \) (where \( x > 0 \))
- Integral of Exponential: \( \int e^x \, dx = e^x + C \)
Common Differential Equations
Differential Equations:
- First-Order Linear Differential Equation: \( \frac{{dy}}{{dx}} + P(x)y = Q(x) \)
Solution: \( y = e^{-\int P(x) \, dx} \left( \int Q(x) e^{\int P(x) \, dx} \, dx + C \right) \) - Separable Differential Equation: \( \frac{{dy}}{{dx}} = f(x)g(y) \)
Solution: \( \int \frac{{dy}}{{g(y)}} = \int f(x) \, dx + C \) - Second-Order Linear Homogeneous Differential Equation: \( a\frac{{d^2y}}{{dx^2}} + b\frac{{dy}}{{dx}} + cy = 0 \)
Solution: \( y = e^{rx} \) where \( r \) is a solution to the characteristic equation \( ar^2 + br + c = 0 \) - Second-Order Linear Non-Homogeneous Differential Equation: \( a\frac{{d^2y}}{{dx^2}} + b\frac{{dy}}{{dx}} + cy = f(x) \)
Solution: \( y = y_h + y_p \) where \( y_h \) is the solution to the associated homogeneous equation and \( y_p \) is a particular solution. - Exact Differential Equation: \( M(x, y) \, dx + N(x, y) \, dy = 0 \)
Solution: If \( \frac{{\partial M}}{{\partial y}} = \frac{{\partial N}}{{\partial x}} \), then the solution is given by \( \int M(x, y) \, dx + \int [N(x, y) - \frac{{\partial}} {{\partial y}} \int M(x, y) \, dx] \, dy = C \) - Riccati Differential Equation: \( \frac{{dy}}{{dx}} = f(x) + g(x)y + h(x)y^2 \)
Solution: If \( y_1 \) is a known solution, then a particular solution is given by \( y = y_1 + \frac{{v'(x)}}{{v(x)}} \), where \( v(x) \) satisfies a linear second-order differential equation.
- Bernoulli Differential Equation: \( \frac{{dy}}{{dx}} + P(x)y = Q(x)y^n \)
- Homogeneous Differential Equation: \( \frac{{dy}}{{dx}} = \frac{{f(ax+by)}}{{f(x)}} \)
- Nonlinear Differential Equations:
- First-order: \( \frac{{dy}}{{dx}} = f(x, y) \)
- Second-order: \( \frac{{d^2y}}{{dx^2}} = f(x, y, \frac{{dy}}{{dx}}) \)
- Systems of Differential Equations:
\[ \frac{{dx}}{{dt}} = f(x, y) \]
\[ \frac{{dy}}{{dt}} = g(x, y) \]
- Boundary Value Problems:
Differential equations with boundary conditions specified at more than one point.
- Partial Differential Equations (PDEs):
Equations involving partial derivatives with respect to multiple variables. Examples include the heat equation, wave equation, and Laplace's equation.
- Green's Functions:
Used to solve inhomogeneous differential equations by representing the response to a delta function input.
Applications of Integration
- Area Under a Curve: \( \int_{a}^{b} f(x) \, dx \) represents the area bounded by the curve \( y = f(x) \), the x-axis, and the lines \( x = a \) and \( x = b \).
- Volume of Revolution: \( V = \pi \int_{a}^{b} [f(x)]^2 \, dx \) represents the volume of the solid generated by revolving the curve \( y = f(x) \) about the x-axis between \( x = a \) and \( x = b \).
- Arc Length: \( L = \int_{a}^{b} \sqrt{1 + [f'(x)]^2} \, dx \) represents the length of the curve \( y = f(x) \) between \( x = a \) and \( x = b \).
- Surface Area of Revolution: \( A = 2\pi \int_{a}^{b} f(x) \sqrt{1 + [f'(x)]^2} \, dx \) represents the surface area of the solid generated by revolving the curve \( y = f(x) \) about the x-axis between \( x = a \) and \( x = b \).
- Work: \( W = \int_{a}^{b} F(x) \, dx \) represents the work done by a force \( F(x) \) moving an object along the x-axis from \( x = a \) to \( x = b \).
- Center of Mass: \( x_{\text{cm}} = \frac{{\int_{a}^{b} x \cdot w(x) \, dx}}{{\int_{a}^{b} w(x) \, dx}} \) represents the x-coordinate of the center of mass of a one-dimensional object with variable density \( w(x) \) between \( x = a \) and \( x = b \).
- Probability: \( P(a \leq X \leq b) = \int_{a}^{b} f(x) \, dx \) represents the probability that a continuous random variable \( X \) lies between \( a \) and \( b \).
- Electric Charge: \( Q = \int_S \rho(x, y, z) \, dV \) represents the total electric charge contained within a three-dimensional region \( S \) with charge density \( \rho(x, y, z) \).
Infinite Sequences and Series
Infinite Sequences:
- An infinite sequence is a list of numbers written in a definite order: \( a_1, a_2, a_3, \ldots, a_n, \ldots \)
- The \(n\)th term of the sequence is denoted as \(a_n\).
- A sequence can be defined explicitly by a formula, recursively, or by some other rule.
Infinite Series:
- An infinite series is the sum of the terms of an infinite sequence: \( \sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + \ldots + a_n + \ldots \)
- The sum of the first \(n\) terms of the series is denoted as \(S_n\).
- If the sequence of partial sums \(S_1, S_2, S_3, \ldots, S_n, \ldots\) converges to a finite number \(S\), the series is said to converge, and \(S\) is called the sum of the series.
- If the sequence of partial sums does not converge, the series is said to diverge.
- Common types of series include arithmetic series, geometric series, telescoping series, and power series.
Parametric Equations and Polar Coordinates
Parametric Equations:
- Parametric equations are a set of equations that express a set of quantities as explicit functions of a number of independent variables, known as parameters.
- Parametric equations are often used to describe curves and paths in the plane or space.
- A parametric curve is defined by two functions \( x = f(t) \) and \( y = g(t) \), where \( t \) is the parameter.
- The coordinates \( (x, y) \) of points on the curve are determined by plugging values of \( t \) into the parametric equations.
- Velocity and Acceleration: Given parametric equations \( x = f(t) \) and \( y = g(t) \), the velocity vector \( \mathbf{v} = \left[\frac{{dx}}{{dt}}, \frac{{dy}}{{dt}}\right] \) and acceleration vector \( \mathbf{a} = \left[\frac{{d^2x}}{{dt^2}}, \frac{{d^2y}}{{dt^2}}\right] \) can be calculated by differentiating the position functions with respect to time.
- Arc Length: The arc length of a curve defined by parametric equations \( x = f(t) \) and \( y = g(t) \) over the interval \( [a, b] \) is given by: \[ L = \int_{a}^{b} \sqrt{\left(\frac{{dx}}{{dt}}\right)^2 + \left(\frac{{dy}}{{dt}}\right)^2} \, dt \]
- Surface Area of Revolution: The surface area generated by revolving the curve defined by parametric equations \( x = f(t) \) and \( y = g(t) \) about the x-axis or y-axis can be calculated using appropriate integration formulas.
Polar Coordinates:
- Polar coordinates are a system of coordinates used to locate a point in a plane by its distance from a fixed point (the pole) and the angle from a fixed direction (the polar axis or polar angle).
- A point in the plane is represented by the polar coordinates \( (r, \theta) \), where \( r \) is the distance from the pole and \( \theta \) is the angle measured counterclockwise from the polar axis.
- Converting between polar and rectangular coordinates: \[ x = r \cos(\theta) \] \[ y = r \sin(\theta) \]
- Common shapes described by polar coordinates include circles, cardioids, limaçons, and roses.
- Area in Polar Coordinates: The area enclosed by a curve described in polar coordinates is given by: \[ A = \frac{1}{2} \int_{\alpha}^{\beta} [f(\theta)]^2 \, d\theta \] where \( \alpha \) and \( \beta \) are the angles corresponding to the starting and ending points of the curve.
- Arc Length: The arc length of a curve defined in polar coordinates \( r = f(\theta) \) over the interval \( [\alpha, \beta] \) is given by: \[ L = \int_{\alpha}^{\beta} \sqrt{[f'(\theta)]^2 + [f(\theta)]^2} \, d\theta \]
Vectors and the Geometry of Space
Vectors:
- A vector is a mathematical object that has both magnitude and direction.
- In three-dimensional space, a vector is often represented as an ordered triple \( \mathbf{v} = \langle a, b, c \rangle \).
- Vector Addition: \( \mathbf{v} + \mathbf{w} = \langle v_1 + w_1, v_2 + w_2, v_3 + w_3 \rangle \)
- Scalar Multiplication: \( c\mathbf{v} = \langle cv_1, cv_2, cv_3 \rangle \) where \( c \) is a scalar.
- Dot Product: \( \mathbf{v} \cdot \mathbf{w} = v_1w_1 + v_2w_2 + v_3w_3 \)
- Cross Product: \( \mathbf{v} \times \mathbf{w} = \langle v_2w_3 - v_3w_2, v_3w_1 - v_1w_3, v_1w_2 - v_2w_1 \rangle \)
Geometry of Space:
- Lines and Planes: Equations of lines and planes can be expressed in vector form using a point on the line/plane and a direction vector.
- Distance Formula: The distance \( d \) between two points \( P(x_1, y_1, z_1) \) and \( Q(x_2, y_2, z_2) \) in space is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
- Midpoint Formula: The midpoint \( M \) of the line segment joining two points \( P(x_1, y_1, z_1) \) and \( Q(x_2, y_2, z_2) \) is given by: \[ M\left(\frac{{x_1 + x_2}}{2}, \frac{{y_1 + y_2}}{2}, \frac{{z_1 + z_2}}{2}\right) \]
- Equation of a Sphere: The equation of a sphere with center \( (h, k, l) \) and radius \( r \) is given by: \[ (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 \]
Vector-Valued Functions
Definition:
- A vector-valued function, also known as a vector function, maps each value of a parameter to a corresponding vector.
- For example, a vector-valued function in two dimensions can be written as \( \mathbf{r}(t) = \langle f(t), g(t) \rangle \).
- In three dimensions, it can be written as \( \mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle \).
Properties:
- Vector Addition: If \( \mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle \) and \( \mathbf{s}(t) = \langle p(t), q(t), r(t) \rangle \) are vector-valued functions, then \( \mathbf{r}(t) + \mathbf{s}(t) = \langle f(t) + p(t), g(t) + q(t), h(t) + r(t) \rangle \).
- Scalar Multiplication: If \( \mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle \) is a vector-valued function and \( c \) is a scalar, then \( c\mathbf{r}(t) = \langle cf(t), cg(t), ch(t) \rangle \).
Additional Formulas:
- Differentiation: If \( \mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle \) is a vector-valued function, then the derivative \( \mathbf{r}'(t) \) is given by: \[ \mathbf{r}'(t) = \langle f'(t), g'(t), h'(t) \rangle \]
- Arc Length: The arc length of a curve defined by a vector-valued function \( \mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle \) over the interval \( [a, b] \) is given by: \[ L = \int_{a}^{b} |\mathbf{r}'(t)| \, dt = \int_{a}^{b} \sqrt{(f'(t))^2 + (g'(t))^2 + (h'(t))^2} \, dt \]
- Curvature: The curvature \( \kappa \) of a curve defined by a vector-valued function \( \mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle \) is given by: \[ \kappa = \frac{|\mathbf{r}'(t) \times \mathbf{r}''(t)|}{|\mathbf{r}'(t)|^3} \]
- Tangent, Normal, and Binormal Vectors: Given a curve defined by \( \mathbf{r}(t) \), the tangent vector \( \mathbf{T} \), normal vector \( \mathbf{N} \), and binormal vector \( \mathbf{B} \) are defined as follows: \[ \mathbf{T} = \frac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|} \] \[ \mathbf{N} = \frac{\mathbf{T}'(t)}{|\mathbf{T}'(t)|} \] \[ \mathbf{B} = \mathbf{T} \times \mathbf{N} \]
Applications:
- Vector-valued functions are often used to describe the path of a particle in space.
- They are also used in physics to represent quantities such as velocity and acceleration, which are vector quantities.
- Vector calculus involves operations and concepts related to vector-valued functions, such as differentiation, integration, and line integrals.
Integration Techniques and Applications
Integration Techniques:
- Integration by Parts: \( \int u \, dv = uv - \int v \, du \)
- Trigonometric Integrals: Various trigonometric identities are used to simplify integrals involving trigonometric functions.
- Integration by Substitution: \( \int f(g(x)) \cdot g'(x) \, dx = \int f(u) \, du \), where \( u = g(x) \).
- Partial Fraction Decomposition: Used to decompose a rational function into simpler fractions for easier integration.
- Improper Integrals: Integrals where the limits of integration are infinite or the function has an infinite discontinuity.
- Integration by Partial Fractions: Used to decompose rational functions into simpler fractions for easier integration.
- Trigonometric Substitution: Used to simplify integrals involving radicals of trigonometric functions.
Applications of Integration:
- Area Under a Curve: Integration is used to find the area enclosed by curves in the plane.
- Volume of Solids of Revolution: Integration can determine the volume of a solid obtained by revolving a region about an axis.
- Work and Fluid Forces: Integration is used to calculate work done by a force and fluid forces on submerged surfaces.
- Center of Mass and Moments of Inertia: Integration helps in finding the center of mass and moments of inertia of objects.
- Probability and Statistics: Integration is used to compute probabilities and expected values in probability theory and statistics.
- Arc Length: Integration is used to find the length of a curve defined by a function \(y = f(x)\) over an interval \([a, b]\): \[ L = \int_{a}^{b} \sqrt{1 + (f'(x))^2} \, dx \]
- Surface Area of Revolution: Integration can find the surface area generated by revolving a curve around the x-axis or y-axis.
- Center of Mass: Integration helps determine the center of mass of an object by considering the distribution of mass.
- Moments of Inertia: Integration is used to calculate moments of inertia, which quantify an object's resistance to rotational motion.
- Electric Charge and Flux: In physics, integration is used to calculate electric charge distribution and electric flux through surfaces.
- Heat Transfer and Thermodynamics: Integration is applied to solve problems related to heat transfer, such as finding the amount of heat transferred or temperature distribution.
Taylor and Maclaurin Series
Taylor Series:
- The Taylor series of a function \( f(x) \) centered at \( x = a \) is given by: \[ f(x) = \sum_{n=0}^{\infty} \frac{{f^{(n)}(a)}}{{n!}} (x - a)^n \]
- It represents the function as an infinite sum of terms involving derivatives of \( f(x) \) evaluated at \( x = a \).
- The Taylor series provides an approximation of the function around the point \( x = a \).
Maclaurin Series:
- The Maclaurin series is a special case of the Taylor series, where \( a = 0 \).
- For a function \( f(x) \), its Maclaurin series is given by: \[ f(x) = \sum_{n=0}^{\infty} \frac{{f^{(n)}(0)}}{{n!}} x^n \]
- It represents the function as an infinite sum of terms involving derivatives of \( f(x) \) evaluated at \( x = 0 \).
- Many common functions have well-known Maclaurin series expansions, such as \( e^x \), \( \sin(x) \), \( \cos(x) \), and \( \ln(1 + x) \).