MTH 133: Calculus II
MTH 133: Calculus II
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NOTES BY SECTION:
Volumes: This section teaches us how to find the volume of solids of revolution and the volume of shapes whose cross-sectional area is known.
Work: The section focuses on a concept from physics that is basic application of integral calculus.
Natural Log Function: A logarithm is an inverse to an exponential. When the base of the logarithm is Euler's number e, the logarithm is called natural.
Natural Exponential Function: The inverse to the natural logarithm is given by the exponential function. It's derivative and integral are perhaps the nicest in all of calculus.
General Log and Exponential Functions: Expanding on some ideas from the past two sections in a more general setting.
Separable Differential Equations: Differential equations (DEs) allow you to write many of nature's phenomenons into single equations. A separable DE is one where we are able to move everything with x to one side, everything with y to the other and compute two integrals to get our solution.
Natural Growth and Decay: Using the language of natural logs and exponentials along with separable DEs, we solve some real-world problems.
Inverse Trig Functions: The input to a trig function is an angle (for this class, in radians) and the output is a ratio of sides of a right triangle containing the angle we input. However, sometimes we only have information about the side lengths of a right triangule and want to figure out what angles it has. This is where the inverse trig functions come into play.
L'Hôpital's Rule: This is maybe every calculus student's favorite way to deal with limits. Be careful, though as it can only be used in certain circumstances.
Integration by Parts: Using the product rule for derivatives, one can obtain a "product rule" for integrals.
Trig Integrals: This section teaches us how to integrate a wide variety of functions which are products or transformations of basic trigonometric functions.
Trig Sub: A very powerful integration technique using substitution that turns certain rational functions into trig integrals. From there, the methods from last section can be applied.
Partial fraction decomposition: An algebraic technique that can reduce general rational functions into sums of simpler rational functions whose integral is typically more approachable.
Sequences: Infinite lists of real numbers. We are interested in what happens as you get "very far out" into the list.
Series: Taking an infinite sum of a sequence. We are interested in which sequences give a sum that "makes sense." That is, what sequences give rise to infinite sums that we can concretely assign a value to? This will allow us to make sense of infinite addition of real numbers.
Integral Test: A way to determine convergence of a series by taking an integral.
Comparison Tests: A way to determine convergence or divergence of a series using some other one whose behavior is known.
Alternating Series: Series asscoiated to sequences whose sign changes each term: +,-,+,-,...
Ratio Test: An extremely useful convergence test, especially when exponents and factorials are present.
Power Series: Polynomials with infinitely many terms. These functions can be used to approximate other functions and encode information about sequences.
More Power Series: We discuss taking derivatives and integrals of power series.
Taylor and Maclaurin Series: When a power series is used to represent a certain function around some point, we get what's known as a Taylor series for that function. A Maclaurin series is just a Taylor series when the point we are approximating our function around is 0.
Parametric Curves: Curves with direction that can be used to represent an object moving around the xy-plane over some time interval.
Calculus + Parametric Equations: Applying ideas from calculus to parametric curves.
Arc Length: We use calculus to find the length of curves described by functions of the form y=f(x).
Polar Coordinates: These coordinates give us a new way to describe points in the xy-plane. Under this coordinate system, circular curves take on much simpler equations.
Areas and Lengths in Polar Coordinates: Transferring the language of calculus over to polar coordinates, we are able to find areas and lengths in a polar coordinate system.
SOME DESMOS GRAPHS: