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Calculus

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Convergence
Integral TestIf f is a continuous function, it converges if and only if its integral also converges
P-series Test\(\sum_{n=1}^\infty n^{-p}\) converges for all p > 1
Comparison TestIf a(n) is convergent and is always bigger than b(n) in an appropriate range, b(n) is also convergent.
If a(n) is divergent and is always smaller than b(n) in an appropriate range, b(n) is also divergent
Limit Comparison TestFor a(n) and b(n), where i and j are their respective limits towards infinity, if i/j = c > 0 and is finite, then both functions converge or diverge (same behaviour).
Alternating Series TestAn alternating series is one where the terms switch signs for every adjacent term. The series converges if its sequence in absolute values is decreasing and if it approaches 0 as \(n \rightarrow \infty\)
Ratio TestFor \(\sum a_n\), define \(L = \lim \limits_{n \to \infty} \lvert \frac{a_{n+1}}{a_n} \rvert \)
If L < 1: series is absolutely convergent
If L > 1: series is divergent
If L = 1: need further testing
Used to find interval of convergence
Root TestFor \(\sum a_n\), define \(L = \lim \limits_{n \to \infty} \sqrt[n]{\lvert a_n \rvert} \)
If L < 1: series is absolutely convergent
If L > 1: series is divergent
If L = 1: need further testing
Finding Power Series
  • See Power Series List above
  • Use integrals and derivatives to convert from known series
  • Replace x with desired variable
MacLaurin's FormulaLet \(f^{(m)}\) be the mth derivative of f(x)
\(f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!} * x^n \)
If you stop at m — 1, \( R_m(x) = \frac{f^{[m]}(c) * x^n}{m!}, 0 \le c \le x\)
Taylor Series\( f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n \)
Arc Length\( \int \sqrt{1 + f'(x)^2} dx \\ \int \sqrt{{x'}^2 + {y'}^2 + {z'}^2} dt = \int ||r'(t)||dt \)
Vectors
\( a \cdot b \)\( a_1b_1 + a_2b_2 + a_3b_3 + ... + a_nb_n \\ a \cdot b = |a||b|cos\theta \)
\( a \times b \\ a = \langle a_1, a_2, a_3\rangle\,and\,b = \langle b_1, b_2, b_3\rangle \\ \text{* Only for 3D vectors} \)\( \langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2, a_2b_1\rangle \\ = |a||b|sin\theta \)
\( |\overrightarrow{a}| \)\( \sqrt{{a_1}^2 + {a_2}^2 + {a_3}^2 + ...} \)
\( proj_ab \)\( \dfrac{a \cdot b}{|a|^2} a \)
\( \text{Area of Parallelpiped} \)\( V = |a \cdot (b \times c)| \\ \text{If V = 0, a, b, and c are coplanar} \)
\( \text{T (unit tangent vector)} \)\( \dfrac{r'(t)}{|r'(t)|} \)
\( \kappa \text{ (Curvature)} \)\( \left|\dfrac{dT}{ds}\right| \\ \kappa(t) = \dfrac{|T'(t)|}{|r'(t)|} = \dfrac{|r'(t) \times r''(t)|}{{|r'(t)|}^3} \\ \text{When y = f(x)} \quad \kappa(x) = \dfrac{|f''(x)|}{[1 + (f'(x))^2]^{3/2}} \)
\( N(t) \text{ (Normal Vector)} \)\( \dfrac{T'(t)}{|T'(t)|} = B(t) \times T(t) \)
\( B(t) \text{ (Binormal Vector)} \)\( T(t) \times N(t) = \dfrac{r'(t) \times r''(t)}{|r'(t) \times r''(t)|} \)
\( v(t) \)\( r'(t) \)
\( a \)\( v'T + \kappa{v}^2N \\ a_TT + a_NN \text{ (see below)} \)
\( a_T \text{ (Tangential acceleration)} \)\( \dfrac{r'(t) \cdot r''(t)}{|r'(t)|} \)
\( a_N \text{ (Normal acceleration)} \)\( \dfrac{r'(t) \times r''(t)}{|r'(t)|} \)
\( (f_x)_y \\ \text{Higher Derivatives} \)\( f_{xy} = \dfrac{\partial}{\partial{y}}\left(\dfrac{\partial}{\partial{x}}\right) = \dfrac{\partial^2f}{\partial{y}\partial{x}} = \dfrac{\partial^2z}{\partial{y}\partial{x}} \)
\( \text{Tangent plane to surface of z} \)\( z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0) \)
\( \text{Total Differential} \)\( d_z = f_x(x, y)dx + f_y(x, y)dy = \dfrac{\partial{z}}{\partial{x}}dx + \dfrac{\partial{z}}{\partial{y}}dy \)
\( \dfrac{\partial{u}}{\partial{t_i}} \\ \text{u is a differential function of n variables} \)\( \dfrac{\partial{u}}{\partial{t_i}} = \dfrac{\partial{u}}{\partial{x_1}}\dfrac{\partial{x_1}}{\partial{t_i}} + \dfrac{\partial{u}}{\partial{x_2}}\dfrac{\partial{x_2}}{\partial{t_i}} + ... + \dfrac{\partial{u}}{\partial{x_n}}\dfrac{\partial{x_n}}{\partial{t_i}} \)
\( \nabla f(x, y) \text{ (gradient of f)} \)\( \langle f_x(x, y), f_y(x, y) \rangle = \dfrac{\partial f}{\partial x}i + \dfrac{\partial f}{\partial y}j \)
\( D_uf(x, y, z) \text{ (directional derivative)} \)\( \nabla f \cdot u \\ \text{Maximum value of directional derivative is } |\nabla f| \)
\( \text{Second Derivative Test} \)\( D = D(a, b) = f_{xx}(a, b)f_{yy}(a, b) - [f_{xy}(a, b)]^2 \\ \text{If D > 0 and $f_{xx}(a, b)$ > 0, then $f(a, b)$ is a local minimum} \\ \text{If D > 0 and $f_{xx}(a, b)$ < 0, then $f(a, b)$ is a local maximum} \\ \text{If D < 0, then $f(a, b)$ is not a local maximum or minimum (saddle point)} \)
\( \text{Lagrange Multipliers} \)\( \text{Find all values of x, y, z, and $\lambda$ such that $\nabla f(x, y, z) = \lambda \nabla g(x, y, z)$ and } g(x, y, z) = k \\ \text{Evaluate $f$ at all points (x, y, z) from the values above; the largest is the maximum and the smallest is the minimum} \)
\( \text{V of solid above rectangle R and below } \)
\( \int \int_D f(x, y)dA \\ f \text{ is continuous on } \\ D = \{(x, y) \,|\, a \le x \le b,\, g_1(x) \le y \le g_2(x)\} \)\( \int^b_a \int^{g_2(x)}_{g_1(x)} f(x, y)dydx = \int^{g_2(x)}_{g_1(x)} \int^b_a f(x, y)dxdy \\ \text{When D is a rectangle, $g_1(x)$ and $g_2(x)$ are constants} \)