Calculus vs precalculus: all you need to know about

For students wondering if calculus is harder than precalculus, this post is for you. In this post, you will:

• discover which is harder, calculus or precalculus
• what students study in calculus and precalculus
• give some tips and advice for students who are taking classes in precalculus and calculus

So keep reading to not miss steps and prevent falling into many mistakes that a lot of students make.

Which is harder calculus or precalculus?

Calculus is harder than precalculus, is nothing to say especially if we compare precalculus to calculus 3. To explain more, precalculus is just an introduction to some subjects you will use in calculus. In precalculus courses, they don’t teach derivatives and integrals which are the hardest and most important foundations of calculus.

In precalculus, they just prepare a student to go and take real calculus courses such as:

• limits
• derivatives
• integrals

Derivatives and integrals are what make calculus harder than precalculus. So for people who find precalculus hard, you haven’t even started the math yet.

this is not offending but just remanding the next stages awaiting you in Calculus 1, 2, and 3.

Is Calculus 1 harder than Precalculus?

Calculus 1 is harder than precalculus because it contains derivatives and integral subjects that aren’t mostly included in precalculus. In precalculus, students don’t study derivatives and integrals. They just study limits. While in Calculus 1 you will study limits, derivatives, and integrals the 3 principal subjects that calculus is based on.

Precalculus is a stage or an introduction that prepares a student for calculus, 1 in the next paragraph we’re going to talk in detail about what students study in precalculus.

So keep reading to know the difference between precalculus and calculus.

What do students study in precalculus?

In precalculus, students study 9 principal subjects, you can find one more or less, but we will mention the principal ones.

1 – Trigonometry

Trigonometry is the study of relationships between the sides and angles of triangles. It can be used to solve triangle problems or find unknown angles or sides of a triangle is given and other applications. 

In precalculus, trigonometry is used to help solve problems that involve quadratic equations and other more complex functions.

Precalculus students typically study the basic concepts of trigonometry, including:

angles in radians and degrees, 

trigonometric functions like cos sin and tang

For further information, you can read this explainable article. 

Trigonometry vs precalculus which is harder?

2 – Complex numbers

Complex numbers have applications in many scientific areas, including signal processing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography, and vibration analysis.

Electrical engineers use complex numbers to calculate the impedance or resistance of an electrical circuit to design their electrical circuits correctly.

In complex numbers, you will deal with the real and imaginary numbers as you can from the image below, also making a lot of operations on them.

z= x+iy

x is a real number 

y is an imaginary number

we’re not going to dig into details. We just want to give a close idea about this subject.

3 – Rational functions

In precalculus, you will study how to make an operation to these rational equations like how to:

• reduce rational equation
• discontinuity of rational functions
• plot a graph of rational equation
• making applications to these rational equalizations like addition, subtraction, and multiplication

4 – Conic sections

conic sections are principally 4 forms that are:

• circle
• hyperbole
• parabola
• hyperbole

In the precalculus phase, you will just discover and learn some basics about these conics sections. You won’t dig deeper into the details until you reach calculus.

you can read this article Trigonometry vs Calculus which is easier?

5 – Vectors

A vector is a mathematical object that has both a magnitude and a direction. Vectors can be used to represent physical quantities such as force, velocity, or displacement. For example, imagine you are pushing a box across the room. The magnitude of the force you are exerting is the amount of force you are using, and the direction is the direction in which the box is moving.

watching this video will be useful to learn more about vectors.

6 – Matrices

In mathematics, matrices are very important in real-life applications like electric and quantum mechanics, matrices are in this form below

in the subject of matrices, you will learn

• Using matrices to represent data: 
• solve equations using matrices
• Matrices
• solving a linear system with matrices

understanding matrices is very important, and having a solid background in them is very required.

you can read this article 4 important steps to learn math from the ground up

7 – Probability and combinatorics

probability and combinatory math are like playing a lottery but using science. In this subject, the principal thing is to know how to transform proper even numerical data.

In other words, being able to understand well the problem and apply the right rules to finally come up with the right predictions.

8 – Series

In the precalculus subject, you will study series, especially geometric series, this subject is used a lot in the banking domain.

In addition, students will be discovering two interesting types of series, numeral and geometric series.

A numerical series is a set of numbers that are put in order and have a pattern. An example of a numerical series is the counting numbers: 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20. So as we can se from each step we add by 2.

A geometric series is a type of sequence in math where each number is the result of multiplying the previous number by a certain number. An example would be 1, 2, 4, 8, 16. So in this case we multiplied by 2. 

9 – Limits and continuity

The Limits function is a mathematical tool that can be used to help determine the behavior of a function as it gets closer and closer to a specific number. This can be helpful in understanding things like rates of change or areas under curves. 

For example, if you want to know how fast a car is going when it’s just about to hit 100 mph, you could use the Limits function to help figure that out.

What do students study in calculus?

in calculus, students base on the precalculus subjects and go deeper and harder, calculus can be divided into 3 sections:

• calculus 1
• calculus 2
• calculus 3

Calculus 1

in calculus 1 you will discover new topics such as derivatives and integrals which are really important. To summarize understanding derivatives and integrals is crucial in calculus 1.

In general, the subjects that you will study in calculus1 are:

• Limits and continuity
• Derivatives
• Analyzing functions
• Mean value theorem
• Extreme value theorem and critical points
• Intervals on which a function is increasing or decreasing
• Integrals
• Differential equations

Calculus 2

in calculus 2 you won’t take a new subject like calculus 1. You will just go deeper about derivatives and integrals that you have studied before in Calculus 1.

• Integrals review
• Integration techniques
• Differential equations
• Applications of integrals
• Parametric equations, polar coordinates, and vector-valued functions
• Series

At this moment things become hard, but the calculus class or course, that students struggle with is the last one that we’re going to talk about in the next paragraph.

Calculus 3

Calculus 3 is the hardest math subject in mathematics, calculus 3 is nothing to compare between precalculus, calculus 1, or even 2. In Calculus 3 the gears of your mind will start making sounds, this is not to offend you. But as an encouragement to put double effort to understand these subjects.

in calculus 3 you will study multivariable which means functions that have multi variables like in the example below

F(x,y,z)

you will need to make derivatives of these multivariable functions, this subject is called partial differentiation also:

• Integrating multivariable functions
• studying vector fields
• discover Green’s, Stokes’,
• know what is the divergence theorems

theses subjects are the toughest subject in calculus, they are the last stage.

In other words, they require a solid base in precalculus and the previous calculus classes 1 and 2. If you don’t have a solid base in these models you will beautifully fail.

Conclusion

calculus is sequentially, So to understand :

• Calculus 1, you need to understand precalculus
• Calculus 2 you need to understand Calculus 1
• Calculus 3 you need to understand Calculus 2

it is like a ladder that you should climb, So you have to begin from the bottom.