Q4 BASIC CALCULUS REVIEWER
S.Y. 2024 - 2025 | D.O. Modules 1 - 18
Integration by Substitution Yielding to
Illustration of an Antiderivative of a Function Inverse Circular Functions
Antidifferentiation - the process of finding the
antiderivative of a given function.
The symbol ∫ is the integral sign that denotes
the operation of antidifferentiation and the
function f is called the integrand.
If F is an antiderivative of f, we write:
∫ f(x)dx = F(x) + C.
Antiderivatives of Polynomial and
Radical Functions
Application of Antidifferentiation to
Differential Equations
A differential equation (DE) is an equation that
involves x, y, and the derivatives of y.
The order of a differential equation pertains to
the highest order of the derivative that appears
in the differential equation.
Antiderivatives of Exponential & Logarithmic
Functions A solution to a differential equation is a function
y = f(x) or a relation f(x, y) = 0 that satisfies the
equation.
Solving a differential equation means finding
all possible solutions to the DE.
Antiderivatives of Trigonometric Functions If f(x) and g(y) are functions in terms of x and y,
respectively, then
𝑑𝑦
𝑔(𝑦) 𝑑𝑥 = 𝑓(𝑥)
Which is a first-order differential equation that
can be solved using a technique called
separation of variables. Thus, it is called a
Antidifferentiation by Substitution separable differential equation. Usually, the
general solution of a separable differential
Integration by Substitution also called equation requires putting all the terms involving
“u-Substitution” is a method to find an integral, the variable y on the left side and those
but only when it can be set up in a special way. involving the variable x on the right side.
The first and most vital step is to be able to Then, the equation is integrated on both sides.
𝑑𝑦
write our integral in this form. 𝑔(𝑦) = 𝑓(𝑥) ⇔ 𝑔(𝑦)𝑑𝑦 = 𝑓(𝑥)𝑑𝑥
𝑑𝑥
⇔ ∫𝑔(𝑦)𝑑𝑦 = ∫𝑓(𝑥)𝑑𝑥
∫𝑓(𝑔(𝑥)) 𝑔'(𝑥)𝑑𝑥
If there are initial conditions, or if we know that
the solution passes through a point, we can
solve this constant and get a particular solution
to the differential equation.
When a problem involves finding a particular Examples:
solution to the differential equation, i.e., a
function y of x given its derivative and its initial
value 𝑦𝑜 at a point 𝑥𝑜 , then we have an initial
value problem.
Application of Differential
Equations in Solving Exponential
Growth and Decay Problems
Let y = f(t) be the size of a certain population at
time t, and let the birth and death rates be the
positive constants b and d, respectively. The
𝑑𝑦
rate of change 𝑑𝑡
in the population y with
respect to the time given by
𝑑𝑦
𝑑𝑡
= 𝑘𝑦 where 𝑘 = 𝑏 − 𝑑
Left, Right, Midpoint Riemann Sums
If k is positive, that is when b > d, then there are
𝑑𝑦 Left Riemann Sums - nth left Riemann sum
more births than deaths and denotes
𝑑𝑡 denoted by (Ln) is the sum of the areas of the
growth. If k is negative, that is when b < d, then rectangles whose heights are the functional
𝑑𝑦
there are more deaths than births and 𝑑𝑡 values of the left endpoints of each subinterval.
denotes decay.
Right Riemann Sums - sum denoted by (Rn) is
Exponential Growth Law and Exponential the sum of the areas of the rectangles whose
Decay Law - states that some quantities grow or heights are the functional values of the Right
decay at a rate proportional to their size. endpoints of each subinterval.
𝑘𝑡
𝑦𝑡 = 𝑦0𝑒 Midpoint Riemann Sums - the nth Midpoint
Riemann sum denoted by (Mn) is the sum of the
areas of the rectangles whose heights are the
Definite Integral as functional values of the Midpoints of the
Limit of Riemann Sums
endpoints of each subinterval. We will use :
𝑥𝑖−1 − 𝑥𝑖
𝑚𝑖 = 2
Fundamental Theorem of Calculus
(Definite Integral)
This theorem states that the derivative of F is f.
In other words, F is an antiderivative of f. Thus,
the theorem relates differential and integral
calculus, and tells us how we can find the area
under a curve using antidifferentiation.
𝑏
∫ 𝑓(𝑥) 𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎)
This sum is called a Riemann Sums for f on [a, 𝑎
b] named after George Bernhard Friedrich
Riemann, a German Mathematician who studied P.S. Refer to the D.O. Modules for examples and
extensively the limits of both continuous and previous quizzes, seatworks, and assignments !
discontinuous function. https://drive.google.com/drive/u/0/folders/1ZPj0hIx9
As the number of rectangles increases, n gets rAR_K_cdyr17zwifgyRtONGJ?fbclid=IwY2xjawImKjlle
larger and larger while the length of the sub HRuA2FlbQIxMQABHTFW5Ta4YvTZSN08WPl7qv11Jf
intervals Δx gets smaller and smaller. 6VmOOxmiFK_7nJAaqgWSsSqlyNxcPE-g_aem_jTrE
MZPUNXEw_psLrt1n1Q
𝑏−𝑎
∆x = 𝑛
is the length of each subintervals. This
is done by dividing equally the [a,b] into n
subintervals.
