Things you should have learned in Calculus II

Transcription

1 Things you should have learned in Calculus II 1 Vectors Given vectors v = v 1, v 2, v 3, u = u 1, u 2, u Common Operations Operations Notation How is it calculated Other Notation Dot Product v u v 1 u 1 + v 2 u 2 + v 2 u 3 Called Inner Product (u, v) and v v = v 2 Addition v + u Magnitude u Also u Cross Product v u The resulting vector is orthogonal to v and u 1.2 Angles The angle found between u and v is given by : u and v are orthogonal if : u and v are parallel if : 1.3 Projections The scalar projection of u onto v is given by: comp v u = In 2D cases, this can be understood as the length of the shadow that u would cast on v The vector projection of u onto v is given by: proj v u = 1.4 Vector Equations The equation of a line: r(t) = r o +tv where r o is and v is A line can also be represented using the parametric equations or symetric equations: The equation of a line segment (NOT THE COMPLETE LINE) is given by r(t) = (1 t)r o + tr 1 where r o is the initial pt and r 1 the terminal pt. To cover the line segmenet t. The equation of a plane is given by where n 1, n 2, n 3 is a vector to the plane and (x o, y o, z o ) is a

2 2 Derivative Techniques 2.1 Integration by Parts The formula for integration by parts : How do we select u? 2.2 Trig substitution How do we know when to use trig sub? Fill in the following table If it looks like this x = dx = x 2 + a 2 x 2 a 2 a 2 x Trig Integrals Write each of the following identities Pythagorean Identities Power Reduction Double Angle Formula sin(2x) =

3 2.4 Partial Fraction Fill the following table in For this type of factor It looks like The decomposition is Linear (x + a) A x + 1 Irreducible Quadratic Repeated Factor For example, write out the general partial fraction decomposition for 1 x 3 (x 2 + 9)(x 1) 2.5 Integrals you need to know x 2 dx sin x dx 1 b + x 2 dx cos x dx tan x dx cos 2 x dx sec x dx cot x dx csc x dx cos x sin x dx

4 3 Application Formulas 3.1 Arc Length Given a function in R 2, the formula for arc length is Notes: Given a parameterized function with x = f(t) and y = g(t) the formula for arc length is Notes: 3.2 Surface Area Given a function f(x) or f(y), the formula for arc length is Notes: Given a parameterized function with x = f(t) and y = g(t) the formula for arc length is Notes:

5 4 Parameterizations 4.1 Derivatives dy dx = d 2 y dx 2 = 4.2 Tangents To find where the tangent lines are horizontal : To find where the tangent lines are vertical : 4.3 Area The area under a parametric curve is equal to 4.4 Polar Coordinates For polar coordinates, r is the distance from the point (0,0) and θ is the angle off the positive x axis To convert, x = y = r = tan θ = Representations of points in polar coordinates are not unique!

6 5 Conics For each of the following, write the general form of the conic, sketch a graph of it, and list any important information you should know Circle Parabola Ellipse Hyperbola

7 6 Hyperbolic Trig Functions For each of the following, list what the function is equal to sinh (x) = cosh(x) = tanh (x) = coth(x) = sech(x) = csch(x) = Important: Show that cosh 2 x sinh 2 x = 1 Find the derivatives of sinh(x), cosh(x), tanh(x)

8 7 Inverse Trig Functions For each of the following, list what the derivative is equal to d dx sin 1 (x) = d dx cos 1 (x) = d dx tan 1 (x)= d dx cot 1 (x)= d dx sec 1 (x) = d dx csc 1 (x) = Find the derivative of sin 1 (x) using the method learned in class

9 8 Indeterminant Form and L Hospital s Rule List the differnt indeterminant forms: How is L Hospital s Rule used? If we have the form 0, what must we do before using L Hospital s? (Illustrate using the example lim x 0 + xlnx ) If we have the form, what must we do before using L Hospital s? (Illustrate using the example lim sec x tan x) x π 2 If we have the form 0 0, 0, 1, what must we do to use L Hospital s? example lim x 0 xx ) + (Illustrate using the

10 9 Series 9.1 Power series To determine if a power series converges, what test do we perform? There are three options we can find for the convergence of power series Option 1: Series converges for all values of x. How do we know? Option 2: Series diverges for all values of x except for x = a. How do we know? Option 3: Series converges for an interval of values. How do we test the end points in this case? 9.2 Finding the power series representation The power series representation of 1 1+x = We try to make a function resemble this to find the power series representation for it 1 For example, find the power series representation of 4 + x We can also integrate or differentiate power series For example, find the power series representation for tan 1 x and that for ln(x + 1)

11 9.3 Taylor Series The general form for a Taylor series is: A Maclaurin series is a Taylor series centered at List the steps to find the Taylor series of a function f(x) Examples you should be able to do: Taylor series of sin(x) centered at x =0 and that of e x centered at x = 0

12 Tests for Convergence Name When? How is it done? How do we tell? Test for divergence Always check first! limn an if the limit is 0, we do another test. If it is not 0, it diverges P-series Look for 1 n p or np Geometric Look for r n or everything to the n power Integral Test

13 Name When? How is it done? How do we tell? Comparison Test Limit Comparison Test Ratio test Root Test