What is the use of Calculus?
The higher you go in math, the less the class will focus on real life applications. The math professors leave that to the science and engineering professors to teach. This might be fore some of the following reasons:
Calculus and higher mathematics topics are very dense, and there just isn't time to teach both concepts, methods, and applications.
Calculus and higher mathematics are not "everyday math" for most people who use it. By this, I mean you won't use calculus to figure out how much change you should have received from the cashier, or calculating the amount of wallpaper you need to buy. Calculus is applied quite often, but in highly specialized settings. If calculus teachers also taught applications, such applications would be useless to most students. By leaving these topics to the science professors, these topics are taught just to students who will use them. In other words, a physicist, an engineer, and a chemist all use calculus, but they do it in different ways.
Mathematicians sometimes pride themselves in how disconnected from application they can be. I've even heard a mathematician say "If there's an application to this, I don't want to know it." (though he did not say that about calculus).
So now you know why you haven't learned applications, here are some:
Calculus is the study of change. Situations in science or engineering where nothing is changing are pretty boring, so we use calculus to study questions that do change. For example: Newton's second law of motion is $$F=ma$$ where $F$ is force, $m$ is mass, and $a$ is acceleration. Though it is not immediately apparent how this equation can be "solved," solutions to this equation describe the motion of an object when the force $F$ is applied it. Acceleration is the change in velocity per change in time: $$a=\frac{dv}{dt}.$$ Velocity, in turn, is the change in position per change in time, so $$a=\frac{d^2x}{dt^2}.$$ Thus, we are left with $$F=m\frac{d^2x}{dt^2}.$$ Often, force itself is a function of $x$, and possible time $t$ , so we have to solve: $$F(x,t)=m\frac{d^2x}{dt^2}.$$ So position in terms of time, $x(t)$, is a function whose second derivative times $m$ is equal to $F(x,t)$. This is called a differential equation, and is somewhat more difficult to solve than an algebraic equation. But by solving such, we can predict how an object will move in time. I have heard differential equations called the "language of physics," since almost every physical situation can be described in terms of a differential equation, and solving these differential equation is key to understanding how the physical situation evolves in time and space.
If we already know how an object moves through time, that is we know $x(t)$, we can find $v(t)$ and $a(t)$ by taking derivatives.
A very important problem in all of science, engineering, and even economics and finance is that of optimization. Figuring out what choices to make to get the best results is very applicable. A company can decide what price to sell their product at to make the most profit. An urban power can decide the best way to set up utilities. A chemist can calculate the optimal amount of reagent to use to complete an experiment. Optimization almost always means calculus because differentiable functions may reach minima or maxima when their derivatives are zero. So the challenge is to take the derivative, set it equal to zero, and solve for the value of $x$ that makes this true.
Calculus also involves an operation which is, in some sense, the opposite of differentiation called integration. You will probably discuss this in your class. Integration is useful when we must find a "total effect" of a constantly changing cause. For instance, a dam can only hold back a certain force of water behind it. The force from the water depends on water pressure. But the water pressure towards the bottom of the dam is more than at the top. So how can we find the total force? Integration is the key.
It can almost be said that multivaribale calculus was developed with the express purpose of studying fluid mechanics and electricity and magnetism.
Calculus is also heavily used in biology. This isn't really my area, so I don't know all the specifics, but I do know there are differential equations used to predict how a population will grow or shrink in time. This can be used to find the optimal amount of animals to hunt without risking extinction.
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...
Just some examples for the topics you mentioned:
- Motion: The standard kinematics equations give $v = \frac{dx}{dt}$ and $a = \frac{dv}{dt} = \frac{d}{dt}(\frac{dx}{dt})$
- Thermodynamics: Energy, pressure, temperature, etc vary with each other, so you'll have to deal with equations like $dE = TdS - PdV + \mu dN$
- Stoichiometry: Amounts of chemical reactants change over time as the reaction takes place, changing the ratios.
- Chemistry: The speed of a chemical reaction changes over time.
- Electricity: For example, electrical induction is described by the equation $\nabla \times E = - \frac{\partial B}{\partial t}$