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Calculus

Differentiation

This course is aimed at teaching concepts, but some advanced mathematics is required. One important skill is to be able to “graphically differentiate” functions. This means identifying the tangent line at a particular point, and finding the slope of the tangent line using

$\text{slope} = \dfrac{\text{rise}}{\text{run}} = \dfrac{\Delta y}{\Delta x}$

The graph below shows an example of graphically differentiating by finding the tangent line (in blue) of the function $$f(x)$$ (in red) at $$x=x_0$$.  It also shows how to calculate the slope of that line by extrapolating the line to $$x = x_1$$ and $$x=x_2$$:

This is one example of how you can graphically differentiate functions, but don't worry about drawing anything like this out. A more important skill is to roughly identify the slope of a tangent line just by looking at the graph. Visualizing images like the one above can be helpful in building this skill.

The table below lists a few functions common in 7C and their derivatives ($$A$$, $$\omega$$, and $$\phi$$ are all constants).

Function $$f(x)$$ Derivative $$f'(x)$$
$$Ax$$ $$A$$
$$A\sin (\omega x + \phi)$$ $$A \omega \cos(\omega x + \phi)$$
$$A\cos (\omega x + \phi)$$ $$-A \omega \sin(\omega x + \phi)$$
$$1/x$$ $$-1/x^2$$

Integration

In this class quantities that accumulate from an initial to a final point can be represented by the area under some curve. For these quantities it's useful to visualize something like the picture below.

An approximation to the area under the curve between $$x_1$$ and $$x_N$$ is given by the area in the shaded rectangles, each with width $$\Delta x$$. As these rectangles become infinitely thin our approximation becomes an exact answer. This process is called integrating the function, and our exact value is the integral of the function.

Useful Integration Identities

In this course, it will be useful to remember these characteristics of integration ($$x_i$$ and $$x_f$$ are arbitrary bounds of integration):

$\int_{x_i}^{x_f} Af(x)\,\mathrm{d}x = A \int_{x_i}^{x_f} f(x)\,\mathrm{d}x \text{ , for any constant } A$

$\int_{x_i}^{x_f} \, \mathrm{d}x = \int_{x_i}^{x_f} (1) \, \mathrm{d}x = \Delta x$

Useful Integral Solutions

Some useful solutions to integrals common to physics classes are shown below (although none of these are essential for 7C)

$\int_{x_i}^{x_f} x\, \mathrm{d}x = \dfrac{1}{2} (x^2_f - x^2_i)$

$\int_{x_i}^{x_f} \dfrac{1}{x^2}\, \mathrm{d}x = - \dfrac{1}{x_f} + \dfrac{1}{x_i}$

$\int_{x_i}^{x_f} \dfrac{1}{x} \, \text{d}x = \ln \left| \dfrac{x_f}{x_i} \right|$

$\int_{x_i}^{x_f} \cos (\omega x + \phi) \, \mathrm{d}x = \dfrac{1}{\omega} \left( \sin(\omega x_f + \phi) - \sin(\omega x_i + \phi) \right)$

$\int_{x_i}^{x_f} \sin (\omega x + \phi) \, \mathrm{d}x =- \dfrac{1}{\omega} \left( \sin(\omega x_f + \phi) - \cos(\omega x_i + \phi) \right)$

These integrals are good for seeing interesting relationships, but the focus of this course is conceptual, not mathematical. Do not spend a lot of time trying to memorize these integrals. Also, if you are still uncertain about any of this material, you should review your calculus notes.