The Mathematics of Calculus: Solutions to Common Problems

Name: The Mathematics of Calculus: Solutions to Common Problems
Rating: 4.5 (17 reviews)

Use Calculus to tackle some mathematical problems

Created byEmanuele Pesaresi

Last updated 1/2025

English

English [Auto],

What you'll learn

How to use the theory of calculus to calculate derivatives
How to use the theory of calculus to calculate definite and indefinite integrals
How to use the theory of calculus to calculate limits
How to use the theory of calculus to calculate volumes of revolution

Course content

4 sections • 23 lectures • 1h 56m total length

Calculation of derivative 15:22
Differentiate a three-term function with sqrt(x) cos(7x) and arcsin(cuberoot(x)) using product and chain rules to obtain the derivative.
Calculation of derivative 28:04
Differentiate a complex function by rewriting it as an exponential, then apply chain rule to the cosine of 2x, the logarithm of one over x squared, and arc cosine terms.
Find the normal to a curve using derivatives7:23
Find the normal to the curve y = e^(1−x^2) via derivatives, then confirm it is perpendicular to the line, yielding the normal equation y = 1/2 x + 1/2.
Derivative of implicit function1:27
Derive the derivative of an implicit parametric function using the chain rule, express dy/dx as (dy/dt)/(dx/dt), and use dx/dt equals one over cosine squared x.
Calculation of the extrema2:57
Analyze derivative to locate critical points at x = 0 and x = 2, study sign changes, and identify f(0) = 0 and f(2) = 12 as maximum and minimum.
Calculation of limits using De L'Hopital1:14
Apply de l'Hopital's rule to evaluate the limit as x approaches zero by differentiating the numerator and denominator. Conclude that the limit equals one sixth.
plot of a function using differential Calculus3:47
Plot the function and note: horizontal asymptote y = 1 as x tends to ±∞, vertical asymptotes at x = ±1, maximum at x = 0 with f(0) = 0.

Definite integral 12:22
Solve a definite integral by substitution: set x-1 = t^2, convert limits from 1 to 5 to 0 to 2, and obtain 4 - 2 arctan(2).
Definite integral 24:38
Derive the indefinite integral of x cos(3x) by rewriting with sin(3x) and cos(3x), then evaluate the definite integral from 1 to π/4.
Calculation of an area1:28
Compute the area under y = ln x between x = e and x = e^2, bounded by y = 0; the integral ∫_e^{e^2} ln x dx equals e^2.
Area in polar coordinates5:40
Compute the area enclosed by r = 2 + cos(2φ) in polar coordinates using the formula area = (1/2) ∫ r^2 dφ from 0 to 2π, yielding 9π/2.
Arc length of a curve9:58
Compute the arc length of y = e^{2x} from x = 0 to x = 1/2 using the line integral dl = sqrt(1+(dy/dx)^2) dx and logarithmic simplifications.
Volume of revolution of a solid4:53
Compute the volume of the solid formed by rotating the region bounded by y = x^2 and y^2 = 2x about the x-axis using washers; the volume is (3/10) π.

Integral 1,2,3,46:34
Compute basic integrals using substitution and recognition: ∫ cos(7x) dx, ∫ x e^{x^2} dx, ∫ (log(x+1))^3/(x+1) dx, and a cotangent-based integral, with results sin(7x)/7 and e^{x^2}/2.
Integral 52:44
Complete the square to rewrite the integral of x over sqrt(5+4x-x^2) as a standard arcsin form, yielding arcsin((x-2)/3) + c.
Integral 64:15
Rewrite the radicand as (x+1)^2+1, use a hyperbolic substitution, and arrive at the inverse hyperbolic sine of x+1 as the integral's result.
Integral 73:00
Compute the integral of the logarithm of x^2+4 divided by x using integration by parts. Derive the result as x log(x^2+4) - 2x + 4 arctan x + C.
Integral 81:11
Solve the integral of x e^{3x} dx by parts, yielding (x e^{3x})/3 - e^{3x}/9 + C.
Integral 91:11
Apply substitution u = tan x to integrate tan^2 x sec^2 x dx, using 1 + tan^2 x = sec^2 x, and obtain tan^3 x / 3 + C.
Integral 108:18
The lecture uses tangent substitution u = tan(x/2) to rewrite sine and cosine, convert the integral to a rational form, decompose into fractions, and express the result with arctan terms.
Integral 112:30
Apply substitution t as the cube root of x to transform the integral, simplify with a division trick, and express the result in terms of t before reverting to x.
Integral 1213:59
Apply partial fraction decomposition to rewrite the integrand as A/(x-3) + B/(x-3)^2 + (Cx+D)/(x^2+1); solve for A,B,C,D and integrate to obtain logs and arctan terms.

Requirements

The prior knowledge requirements are pretty basic. Previous knowledge of the concept of: functions, trigonometry, simple high school algebra and some theory on calculus (especially derivatives and integrals).

Description

In this short course some calculus exercises are solved, in particular on: derivatives, integrals, limits, calculation of areas, arc length, volumes of revolution.

The problems are solved step by step. The prior knowledge requirements are pretty basic. Previous knowledge of the concepts: functions, trigonometry, simple high school algebra would be useful.

In this course Calculus is explained by focusing on understanding the key concepts rather than resorting to rote learning. The process of reasoning by using mathematics is the primary objective of the course, and not simply being able to do computations.

Let's summarize here in the following the two fundamental concepts: differential and integral calculus.

Differential calculus is the study of the definition, properties, and applications of the derivative of a function. The process of finding the derivative is called differentiation. Given a function and a point in the domain, the derivative at that point is a way of encoding the small-scale behavior of the function near that point. By finding the derivative of a function at every point in its domain, it is possible to produce a new function, called the derivative function or just the derivative of the original function.

Integral calculus is the study of the definitions, properties, and applications of two related concepts, the indefinite integral and the definite integral. The process of finding the value of an integral is called integration.

The indefinite integral, also known as the antiderivative, is the inverse operation to the derivative. F is an indefinite integral of f when f is a derivative of F.

The definite integral inputs a function and outputs a number, which gives the algebraic sum of areas between the graph of the input and the x-axis. The technical definition of the definite integral involves the limit of a sum of areas of rectangles, called a Riemann sum.

Finally, let me stress how crucial it is to make efforts while learning. I strongly believe that grasping these topics requires active thinking on your part before the concepts truly sink in. That’s why I often recommend going through the material by doing the calculations yourself (pausing the video if needed). My advice is to approach learning as actively as possible, rather than passively, and be aware that getting stuck sometimes may be beneficial, because it gives you time to think (and while thinking you might even realize whether you are digesting the concepts or not, possibly changing your perspective).

Who this course is for:

Students who want to understand how to tackle calculus exercises using intuition, without resorting to rote learning.

The Mathematics of Calculus: Solutions to Common Problems

What you'll learn

Explore related topics

Course content

Calculation of derivatives7 lectures • 30min

Definite integrals, calculation of areas6 lectures • 29min

Indefinite integrals9 lectures • 44min

Proof of non-trivial trigonometric identity1 lecture • 14min

Requirements

Description

Who this course is for: