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?
Best Answer
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.