# Tips on understanding Calculus?

#### HRubss

So I am currently planning to major in Mechanical Engineering which is heavily involved with Math. I'm taking Calculus 1 this semester and so far I'm doing just above average on all my test. (All B's and 1 A for my limits exam). So far so good right? Eh, I only excel because I just know how to do the problems by memorizing the steps to get the answers, I don't actually 'evaluate' the problems. This is probably because I'm not naturally good at math, I just forced myself to learn the steps. I don't know how to apply these sort of problems to real world scenarios which kind of sucks because math seems really cool when I can actually understand it. Any tips on getting a deeper understanding on math? I would appreciate if any engineer explained his/her day in the life as an engineer and how math comes in to play. (I want to be a ME for car engines and such). Thanks!

#### FactChecker

Gold Member
2018 Award
I used math all the time in studying airplane flight controls. The auto industry has a lot of similar work. I don't know how much it would help to describe a "day in the life", but the airplane linear and rotational accelerations, velocities, and positions are all related to each other in terms of integrals and derivatives. In Calc 1, it may help you a lot to associate acceleration, velocity, and position with integrals and derivatives of each other.

Other than that, keep in mind that there are several integrals and derivatives (involving polynomials, trig functions, exponential, and natural log) that you need to memorize, because no one really derives those after the first time. Then there are a hand full of rules that you will be expected to memorize and be able to use (quotient rule, chain rule, product rule).
PS. My favorite: "If the quotient rule you wish to know, it is low Dhigh less high Dlow, draw a line and down below the denominator squared must go."

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#### Student100

Gold Member
Your course should at sometime talk about specific applications, normally after you've learned the techniques.

I think most applied problems in calculus one are on looking at rate of change, which should be interesting to ME students. An example would be something like jerk, or $\frac {da}{dt}$, which gives you the rate at which acceleration is changing with time. It's probably something structural guys are concerned with, but I have no first hand experince here, I'm guessing.

What I found is the most helpful way to gain a better understanding at this level is if you draw the pictures of what happening. I.e. Draw the derivatives of functions, and when you get to intergrals draw the given function and label the pieces. Then set up things from those drawings. You'd be surprised how many students get to physics, can draw the problem but then have no ideal how to go about setting up the intergral/math or whatever from that picture.

#### chiro

Hey HRubss.

Since calculus deals with finding changes and adding them up, I'd try and focus on understanding how to put those changes into context.

There are many kinds of changes that you look at from lengths, areas, and volumes to things like projections and cross products in vector calculus.

If you understand the nature of the change and what it represents physically then it gets easier to understand what it actually means in some physical way.

The rest is basically algebra, mathematical identities, and language involving proof's and theorems and other mathematical ideas.

#### symbolipoint

Homework Helper
Gold Member
So I am currently planning to major in Mechanical Engineering which is heavily involved with Math. I'm taking Calculus 1 this semester and so far I'm doing just above average on all my test. (All B's and 1 A for my limits exam). So far so good right? Eh, I only excel because I just know how to do the problems by memorizing the steps to get the answers, I don't actually 'evaluate' the problems. This is probably because I'm not naturally good at math, I just forced myself to learn the steps. I don't know how to apply these sort of problems to real world scenarios which kind of sucks because math seems really cool when I can actually understand it. Any tips on getting a deeper understanding on math? I would appreciate if any engineer explained his/her day in the life as an engineer and how math comes in to play. (I want to be a ME for car engines and such). Thanks!
Mathematics of whatever kind one needs for a job (like engineer) must become like a language for him far in advance of becoming an engineer, or taking his first professional job.

Student must study to learn and to understand. Student must not try to rely on just learning what steps to do for solving a problem. Too many possible example variations can happen. No body can memorize all of them and their steps. You must just learn how to think. Students have, in their Algebra, Trigonometry, and Calculus books, applications problems; so this gives ways to begin learning how to think and apply what is being studied.

Both in Mathematics courses and in science or engineering courses, students are supposed to make diagrams, drawings, figures, or /and graphs, label parts with variables and expressions, setup equations or other statements, and solve.

HRubss,
Sometimes a person does not learn a course or a part of a course the first time through. Studying it again either on your own or by taking it again can be a good thing. This can be necessary at some point during ones education. With some, the trouble happens earlier like in Algebra. With other people, the trouble might happen somewhere in Calculus.

#### HRubss

Thanks for the input guys! Any books you guys have to recommend? I know how to do rules such as the product rule, quotient rule, and the chain rule like mentioned above but I don't exactly know why we have to do those rules when solving problems, you know? I can do the quadratic equation but I don't know WHY we have to do it. I hear that many physic laws are proven to be true with math but I have no clue how a Physicist would apply math to the theory in the first place. Maybe I spent too much time learning the steps and not actually understanding why we did those steps? Throughout my years in education, I never actually had a professor prove the steps in an equation, either that or I never really paid attention. I understand how the Pythagorean Theorem is true because I curiously created a real world scenario. I can also see how Pi is derived from a circle as well after yet again, trying a real world scenario (marking and making one complete revolution of tape and measuring the distance with a ruler) but those are the only 2 things I can really understand about math.

#### symbolipoint

Homework Helper
Gold Member
Thanks for the input guys! Any books you guys have to recommend? I know how to do rules such as the product rule, quotient rule, and the chain rule like mentioned above but I don't exactly know why we have to do those rules when solving problems, you know? I can do the quadratic equation but I don't know WHY we have to do it. I hear that many physic laws are proven to be true with math but I have no clue how a Physicist would apply math to the theory in the first place. Maybe I spent too much time learning the steps and not actually understanding why we did those steps? Throughout my years in education, I never actually had a professor prove the steps in an equation, either that or I never really paid attention. I understand how the Pythagorean Theorem is true because I curiously created a real world scenario. I can also see how Pi is derived from a circle as well after yet again, trying a real world scenario (marking and making one complete revolution of tape and measuring the distance with a ruler) but those are the only 2 things I can really understand about math.
That description tells me that you do not understand basic Algebra and you do not have a strong grip on basic Arithmetic. Giving other advice through a forum may be inadequate. Could you find a counselor or adviser at your school to help assess you better?

#### HRubss

That description tells me that you do not understand basic Algebra and you do not have a strong grip on basic Arithmetic. Giving other advice through a forum may be inadequate. Could you find a counselor or adviser at your school to help assess you better?
When you say "don't understand" what do you mean? I can do the work but that's because of memorizing the steps to countless problems. For example, find the x value intercept from this function, y=3x+5. I know to set y=0, subtract the 5 and then divide by 3 to get x alone. x=(5/3). What I don't understand is why we subtracted 5 and then divided by 3. (I know why we set y=0, the y in the ordered pair is 0 whenever it crosses the x-axis) Is there a word of thumb that says to get the variable alone in a function to get it's value? How are these steps naturally correct in our real world? Maybe I'm just thinking too hard.

#### Mark44

Mentor
Thanks for the input guys! Any books you guys have to recommend? I know how to do rules such as the product rule, quotient rule, and the chain rule like mentioned above but I don't exactly know why we have to do those rules when solving problems, you know?
The rules that you mentioned are shortcuts that you use for specialized kinds of functions. The alternative to these shortcuts would be to use the limit definition of the derivative (the limit of the difference quotient, $\lim_{h \to 0}\frac{f(x + h) - f(x)} h$ every time.
If your function involves a product, such as $x^2\cos(x)$, you would use the product rule. If the function involves a quotient, such as $\frac{x - 1}{x + 3}$, you would most likely use the quotient rule. The different rules are tools that let you differentiate different kinds of functions, similar to the way a carpenter uses one tool to saw a board, and another tool to drill a hole.
HRubss said:
I can do the quadratic equation but I don't know WHY we have to do it.
You use the quadratic formula to solve a quadratic equation. It's as simple as that.
HRubss said:
I hear that many physic laws are proven to be true with math but I have no clue how a Physicist would apply math to the theory in the first place. Maybe I spent too much time learning the steps and not actually understanding why we did those steps? Throughout my years in education, I never actually had a professor prove the steps in an equation, either that or I never really paid attention.
Well those are two completely different things. You can't blame your teachers if you're not sure whether you were paying attention.
HRubss said:
I understand how the Pythagorean Theorem is true because I curiously created a real world scenario. I can also see how Pi is derived from a circle as well after yet again, trying a real world scenario (marking and making one complete revolution of tape and measuring the distance with a ruler) but those are the only 2 things I can really understand about math.
When you say "don't understand" what do you mean? I can do the work but that's because of memorizing the steps to countless problems. For example, find the x value intercept from this function, y=3x+5. I know to set y=0, subtract the 5 and then divide by 3 to get x alone. x=(5/3). What I don't understand is why we subtracted 5 and then divided by 3.
You are applying the inverse operations, one at a time, to get x all by itself.
To get x by itself you need to go from 3x + 5 to 3x, and then to x.
To undo the "+ 5" operation, you add -5, but you have to do this to both sides so that the resulting equation is equivalent to the one you started with.
That gets us to y - 5 = 3x
To undo the "X 3" operation (X here means "times"), divide by 3 (or multiply by 1/3, which has the same effect). Again, you have to apply the operation to both sides, which takes us to
(1/3)(y - 5) = x, or y/3 - 5/3 = x
HRubss said:
(I know why we set y=0, the y in the ordered pair is 0 whenever it crosses the x-axis) Is there a word of thumb that says to get the variable alone in a function to get it's value? How are these steps naturally correct in our real world? Maybe I'm just thinking too hard.
There's not a rule of thumb. Different equations require different approaches. Linear equations are pretty easy, but quadratic equation require something extra, Equations with radicals require yet another approach, and so on.

#### HRubss

I guess I really was overthinking these steps. I think I was trying to get a deeper understanding of calculus since I don't really know how I would be able to apply these things in a real world scenario. Maybe that's where physics comes in to play? I think I'm going to take some time and look at proofs of certain equations.

#### symbolipoint

Homework Helper
Gold Member
I guess I really was overthinking these steps. I think I was trying to get a deeper understanding of calculus since I don't really know how I would be able to apply these things in a real world scenario. Maybe that's where physics comes in to play? I think I'm going to take some time and look at proofs of certain equations.
Mark44 in post #9 said best what needs saying. Also, you are on the right track when you begin to believe that Physics will give you ways and whys to apply both your Algebra skills and your Calculus skills. That is one of the things that happen.

#### HRubss

Although my classmates have warned my about how difficult university physics is, I'm kind of eager to take it. I took it back in HS but it was my senior year and at that point I was slacking off but it was an interesting class for sure.

#### analogdesign

I wouldn't worry so much. Very, very few people understand what they are doing when they take calculus. I got As in my calc classes but like you I didn't really feel a connection to the material. You will be MUCH more connected to the material after you start taking engineering classes that use calculus as a means to solve real problems.

I don't think I really understood calculus in any kind of fundamental way until I took my first signal processing courses. Moving back and forth between discrete and continuous time (as well of time and frequency) representations of signals really helped drive the concept of a limit and, for that matter, the fundamental theorem of calculus home for me.

The point is don't sweat it. You'll understand calc much, much better once you get into your thermo and fluid dynamics courses. You'll have no choice.

#### HRubss

Hopefully that comes true, thanks for the notes/tip!

"Tips on understanding Calculus?"

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