These are my complete notes for Integral Calculus & Calculus 2, covering such topics as Definite & Indefinite Integrals, the Fundamental Theorem of Calculus, LRAM, RRAM, and MRAM, the Trapezoid Rule, Exponential Growth/Decay, Trigonometric Antidifferentiation, the Washer Method, Integration by Parts, Power Series & Infinite Series, and more.
I color-coded my notes according to their meaning - for a complete reference for each type of note, see here (also available in the sidebar). All of the knowledge present in these notes has been filtered through my personal explanations for them, the result of my attempts to understand and study them from my classes and online courses. In the unlikely event there are any egregious errors, contact me at jdlacabe@berkeley.edu.
Summary of Integral Calculus & Calculus 2 (Complete)
?. Local Linearity.
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If you continuously zoom in at the value pi/2, you can see the graph flatten until it becomes a line. The line represents the slope of the tangent line at that point. WE can use the tangent line at that point to approximate the value of y = sin(x) for points near x = pi/2. This is called linearization.
# Linear Approximation: If f is differentiable at x=a, then the equation of the tangent line, L(x) = f(a) + f'(a)(x-a), defines the linearization of f at a. The approximation is called the Linear Approximation of f at the point x=a is the center of approximation.
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Finding the Local Linearization is easy as hell. All you need is the base function that will be the f(x), generally the parent function of the value you are trying to approximate such as √x for √51 and cos(x) for cos(1.75), and the value you are using to compare, serving as a.
The formula is L(x) = f(a) + f'(a)(x-a). Usually, they will straight up give you the value they want you to compare to but a lot of the time you will have to make it up for yourself, something like the closest known perfect value associated with the parent function.
The formula is L(x) = f(a) + f'(a)(x-a). Usually, they will straight up give you the value they want you to compare to but a lot of the time you will have to make it up for yourself, something like the closest known perfect value associated with the parent function.
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Derivatives calculate slopes of tanent lines and instantaneous Rates of Change, but in order to describe how these things accumulate over time, we need integral calculus to find the areas under curves.
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RIEMANN SUMS:
The most rudimentary (non-calculus) form of finding the area under a curve on a graph (like y = √x) is as follows:
Partition the area under the curve into vertical strips. If the strips are narrow enough, they are indistinguishable from rectangles, and then by summing all the individual areas of the rectangles you get the area under the curve.
The most rudimentary (non-calculus) form of finding the area under a curve on a graph (like y = √x) is as follows:
Partition the area under the curve into vertical strips. If the strips are narrow enough, they are indistinguishable from rectangles, and then by summing all the individual areas of the rectangles you get the area under the curve.
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A particle's total distance traveled is given by the srea under the VELOCITY curve!!!!!! Change it as needs be.
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Remember that if the rate of change is constant and the function is linear, then you can solve just with geometry: most of the time, using a trapezoid (or rather a bottom rectangle combined with a triangle) will do the trick.
