TITLE: What is the use of Calculus?
QUESTION [9 upvotes]: I know this may seem like a really broad question, but I will narrow it down. I really want to know the purpose of some of the things my teacher is emphasizing in my calc class.
For example why it so important to know:
\begin{align}
\frac{d}{dx}\sin\left(x\right)&=\cos\left(x\right),\\
\frac{d}{dx}\left(\sec\left(x\right)\right)&=\sec\left(x\right)\tan\left(x\right), \text{ or }\\
\lim _{x\to 0}\left(\frac{\sin\left(x\right)}{x}\right)&=1?
\end{align}
I am looking for a real world benefit or application to knowing these things. Yes, its cool to be able to prove things like $\lim\limits _{x\to -\infty \:}\left(\sqrt{4\cdot \:x^2-5\cdot \:x}+2\cdot \:x\right) = \frac{5}{4}$, but how does all these toplics like: End Behavior,Limits, difference quotients, derivatives come into play in real world applications
So, far we learned the difference quotient to find the average rate of change. We learned the end behavior, so we can learn what occurs if a parameter approaches infinity. We learned limits so we can work around computing values at points where the function is not defined. Ex: Dividing by 0. Now we are learning derivatives which is the limit of a difference quotient as $h\to 0$.
But we haven't had any questions yet, that this is used in a practical problem. The questions we are asked in class are purely proofs and computations.
Ex: Find $\frac{d}{dx}\left(\frac{d}{dx}\left(\frac{1}{\sqrt{2\pi \:}}e^{-\frac{x^2}{2}}\right)\right)$
Ex:$$
f(x) = \left\{
        \begin{array}{ll}
            4x^2+1 & \quad x > 2 \\
            17 & \quad x = 2 \\
            16x-15 & \quad x < 2
        \end{array}
    \right.
$$
I asked my teacher this question and he mentioned the purpose of calculus is to find the global and local extrema, finding roots,calculating instantaneous rates of change, but didn't really go into to many real world applications. So far my Calc class has been prove blablabla because you have the math skills to do so. In classes like algebra 2 we didn't just learn algebra, but we learned many real world practical applications for it. My proffessor mentioned Calculus is used a lot in the real world to find area under curves and rates of change. Can you give me some examples on real world applications where I would need to find the area under the curve? Or find the instantaneous rate of change?

REPLY [10 votes]: Somewhat perversely, I choose to respond to precisely your explicit examples.  Mostly because the "surprising thing" is that all of the material you are learning comes from the "Real World".

$\frac{\mathrm{d}}{\mathrm{d}x}\sin(x) = \cos(x)$ -- position and velocity for object in simple harmonic motion.  Protons in the LHC, electrons in a radio antenna, masses dangling from springs, pendula of small displacement, infrared oscillations of molecules -- these are all (approximately) harmonic motions.
$\frac{\mathrm{d}}{\mathrm{d}x}\sec(x) = \sec(x)\tan(x)$ -- rate of change of cost to shingle a roof versus roof angle, or rate of Mercator latitudinal compression versus angle above equator (although the fascinating half of this problem (distortion in Mercator projection) is the secant integral, both discussed at the link.)
$\lim_{x \rightarrow 0} \frac{\sin(x)}{x} = 1$ -- The sinc function is the Fourier transform of a rectangle; the limit you give informs us that rectangular windowing faithfully reproduces the DC component of a signal.
$\frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{\sqrt{2 \pi \,}} \mathrm{e}^{- \frac{x^2}{2}}$ -- this is the second derivative of the normal distribution (a.k.a., Gaussian distribution).  Second derivatives of distributions are commonly used to improve fitting parameters in spectroscopy.
$f(x) = \begin{cases} 4x^2 + 1, &x>2; \\ 17, &x=2; \text{ and} \\ 16x-15, &x < 2 \end{cases}$ -- at low velocities, the drag experienced by nearly spherical particles is linear in their velocity (see Stokes' Law), but at higher velocities, the drag is quadratic in velocity (see Newtonian drag) (with much complexity, because: fluid dynamics over a wide range of Reynolds numbers).  While it is a stretch to say that your example is directly applicable, practice with these simplified model problems is necessary before tackling real examples.

We use integrals (areas under curves) to compute things which accumulate and whose rates we know.  For instance, absorbed dose is an integral of the emission rate of a radiation source.  Light scattering in clouds and fog is modelled by integrals versus penetration depth.  Also, to go from densities to quantities -- mass density to weight, stress-strain to fracture specific energy, pressure to force, surface tension to surface energy, et al...