BC Calculus: Things to Memorize

Basic Derivative and Integral Formulas

• Essential for solving a wide range of calculus problems.

Differentiation Techniques

• Product Rule: Used to differentiate the product of two functions.
  • If $h(x) = f(x)g(x)$, then $h'(x) = f'(x)g(x) + f(x)g'(x)$.
• Quotient Rule: Used to differentiate the quotient of two functions.
  • If $h(x) = \frac{f(x)}{g(x)}$, then $h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$.
• Chain Rule: Used to differentiate composite functions.
  • If $h(x) = f(g(x))$, then $h'(x) = f'(g(x))g'(x)$.

Integration Techniques

• Substitution (u-substitution): Simplifies integrals by changing the variable.
• Integration by Parts: Used to integrate the product of two functions.
  • $\int u dv = uv - \int v du$
• Partial Fractions: Decomposes rational functions into simpler fractions for easier integration.

Mean Value Theorem

• States that if a function $f$ is continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$, then there exists at least one point $c$ in $(a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$.

Intermediate Value Theorem

• States that if $f$ is a continuous function on the closed interval $[a, b]$, and $k$ is any number between $f(a)$ and $f(b)$, then there exists at least one number $c$ in the interval $[a, b]$ such that $f(c) = k$.

Fundamental Theorem of Calculus

• Part 1: If $f$ is a continuous function on $[a, x]$, then the derivative of the integral from $a$ to $x$ of $f(t) dt$ is $f(x)$.
  • $\frac{d}{dx} \int_a^x f(t) dt = f(x)$
• Part 2: The definite integral of $f(x)$ from $a$ to $b$ is the difference of the antiderivative $F(x)$ evaluated at $b$ and $a$.
  • $\int_a^b f(x) dx = F(b) - F(a)$

L'Hopital's Rule

• Used to evaluate limits of indeterminate forms (e.g., $\frac{0}{0}$ or $\frac{\infty}{\infty}$).
• If $\lim<em>{x \to c} \frac{f(x)}{g(x)}$ is of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim</em>{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$, provided the limit exists.

Indeterminate Forms

• Common indeterminate forms include $\frac{0}{0}$, $\frac{\infty}{\infty}$, $0 \cdot \infty$, $\infty - \infty$, $1^\infty$, $0^0$, and $\infty^0$.
• Techniques to handle: L'Hopital's Rule, algebraic manipulation, and rewriting expressions.

Rectangular Approximation and Trapezoidal Rule

• Rectangular Approximation (Riemann Sum): Approximates the area under a curve using rectangles.
• Trapezoidal Rule: Approximates the area under a curve using trapezoids.
  • $\int<em>a^b f(x) dx \approx \frac{\Delta x}{2} [f(x</em>0) + 2f(x<em>1) + 2f(x</em>2) + \dots + 2f(x<em>{n-1}) + f(x</em>n)]$ where $\Delta x = \frac{b-a}{n}$

Volume Formulas

• Disk Method: Used to find the volume of a solid of revolution when the slices are perpendicular to the axis of rotation and form disks.
  • $V = \pi \int_a^b [r(x)]^2 dx$
• Washer Method: Similar to the disk method but used when there is a hole in the solid.
  • $V = \pi \int_a^b ([R(x)]^2 - [r(x)]^2) dx$
• Shell Method: Used when the slices are parallel to the axis of rotation.
  • $V = 2\pi \int_a^b r(x)h(x) dx$

Average Value Formula

• The average value of a function $f(x)$ on the interval $[a, b]$ is given by:
  • $f<em>{avg} = \frac{1}{b-a} \int</em>a^b f(x) dx$

Euler's Method

• A numerical method for approximating the solution of a differential equation.
• Given $\frac{dy}{dx} = f(x, y)$ and an initial condition $y(x<em>0) = y</em>0$, the approximation is:
  • $y<em>{i+1} = y</em>i + f(x<em>i, y</em>i) \Delta x$

Differential Forms of Exponential and Logarithmic Functions

• $\frac{d}{dx}(e^x) = e^x$
• $\frac{d}{dx}(a^x) = a^x \ln(a)$
• $\frac{d}{dx}(\ln x) = \frac{1}{x}$
• $\frac{d}{dx}(\log_a x) = \frac{1}{x \ln(a)}$

Sum of a Geometric Series

• The sum of an infinite geometric series is given by:
  • $S = \frac{a}{1 - r}$, where $a$ is the first term and $r$ is the common ratio, and |r| < 1 for convergence.

Convergence/Divergence Tests

• nth Term Test (Divergence Test): If $\lim<em>{n \to \infty} a</em>n \neq 0$, then the series $\sum a_n$ diverges.
• Geometric Series Test: A geometric series $\sum ar^n$ converges if |r| < 1 and diverges if $|r| \geq 1$.
• Alternating Series Test: If an alternating series satisfies the conditions that the absolute value of the terms decreases monotonically and the limit of the terms is zero, then the series converges.
• Integral Test: If $f(x)$ is a positive, continuous, and decreasing function on $[1, \infty)$, then the series $\sum<em>{n=1}^\infty f(n)$ and the integral $\int</em>1^\infty f(x) dx$ either both converge or both diverge.
• Ratio Test: Let $L = \lim<em>{n \to \infty} |\frac{a</em>{n+1}}{a_n}|$; if $L < 1$, the series converges; if $L > 1$, the series diverges; if $L = 1$, the test is inconclusive.
• Direct Comparison Test: Compare the given series with a known convergent or divergent series.
• Limit Comparison Test: If $\lim<em>{n \to \infty} \frac{a</em>n}{b<em>n} = c$ where $c$ is a finite and positive number, then $\sum a</em>n$ and $\sum b_n$ either both converge or both diverge.

Taylor Series Centered at $x = a$

• The Taylor series of a function $f(x)$ centered at $x = a$ is given by:
  • $\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x - a)^n$

Lagrange Error Bound for Taylor Polynomials

• The error in approximating $f(x)$ by its nth degree Taylor polynomial $P_n(x)$ is bounded by:
  • $|R_n(x)| \leq \frac{M}{(n+1)!}|x - a|^{n+1}$, where $M$ is the maximum value of $|f^{(n+1)}(z)|$ on the interval between $a$ and $x$.

Building Block Maclaurin Series

• $e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$
• $\sin x = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots$
• $\cos x = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \dots$

Arc Length

• For rectangular functions: If $y = f(x)$ on $[a, b]$, then the arc length is:
  • $L = \int_a^b \sqrt{1 + (\frac{dy}{dx})^2} dx$
• For parametric functions: If $x = f(t)$ and $y = g(t)$ on $[a, b]$, then the arc length is:
  • $L = \int_a^b \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} dt$