Introduction to Calculus The course sounds intimidating starting from its name to its content. The first step is the most difficult and challenging and it must be taken care so that we can get all the concepts right. Our plan is to present calculus in a graphical stand point to help students visualize the entire thing. We will also lay the ground work from logic and let us say basic commonsense so that developing the formula wont be so difficult. Lastly present the result with clear and understandable example. The sequence of topics will be as follows: The graphical version of the limit the tangent line differentiation of some algebraic function the fundamental theorem of calculus. Differentiation of trigonometric functions Integration of trigonometric functions Differentiation of logarithmic function Integration of logarithmic functions Differentiation of the natural logarithmic function Integration of the natural logarithmic function Taylors polynomial The chain rule. Differentiation of multi variable function using the chain rule. Integration using the chain rule. the graphical version of limits imagine a line containing infinitely many points. Since it is a line, we do not question ourselves whether it is totally continuous all through out. What if there are instances where some points of this line does not exist? Making the line reach its limit/end. It is as if that point vanished and gone for eternity. Let us examine. Consider the function, f(x)=(x-1)/(x-1) The table below shows the value of f(x). x -5 -4 -3 -2 -1 0 1 2 3 f(x)=(x-1)/(x-1) 1 1 1 1 1 1 0/0 1 1 Results We get an undefined value 0/0 when x is equal to 1 and get a value of f(x)=1 for other values of x. this means that at x = 1 the function is not defined or has no meaning. Also at that point the function is not continuous. Interpretation Limit of a function could mean the last value outputted by f(x) which is defined before it reaches a critical value x→a that will yield the function being discontinuous. lim┬(x→a)⁡〖f(x)=L〗 We can also see that our critical value can be positive or negative hence we can also write the formula for the limit of the function as; lim┬(x→a^+ )⁡〖f(x)=L〗 lim┬(x→a^- )⁡〖f(x)=L〗 Worth to Ask if such a number L, exist then, does that also mean that the function will be discontinuous from that value right away?

1. MATH HUB
Learn math in the easiest way possible!

2. LIMITS
An important concept that make calculus possible!

3. MOTIVATION
•Imagine a line containing infinitely
many points. Since it is a line, we do
not question ourselves whether it is
totally continuous all through out.
What if there are instances where
some points of this line does not
exist? Making the line reach its
limit/end.
Lines!

4. LETS EXAMINE
•Consider the function
•𝑓 𝑥 =
𝑥−1
𝑥−1
Some functions

5. LETS EXAMINE
The table of values
𝑓 𝑥 =
𝑥 − 1
𝑥 − 1
1 1 1 1 1 1 0/0 1 1
𝑥 -5 -4 -3 -2 -1 0 1 2 3

6. LETS EXAMINE
The Graph

7. ANALYSIS • We get an undefined value 0/0 when x is equal to 1 and get
a value of 𝑓 𝑥 = 1 for other values of x. this means that at x
= 1 the function is not defined or has no meaning. Also at
that point the function is not continuous.
Lines!

8. INTERPRETATION
• Limit of a function could mean the last value outputted by
𝑓(𝑥) which is defined before it reaches a critical value when
𝑥 → 𝑎𝑝𝑝𝑟𝑜𝑎𝑐ℎ𝑒𝑠 𝑎 that will yield the function being
discontinuous.
• lim
𝑥→𝑎
𝑓 𝑥 = 𝐿
• We can see that as x approaches the critical value 𝑎 , the
value gets larger and larger and may either be towards the
positive infinity or the negative infinity. Also, apparently x
approaches 𝑎 from positive infinity or negative infinity and
we can write;
• lim
𝑥→𝑎+
𝑓 𝑥 = 𝐿
• lim
𝑥→𝑎−
𝑓 𝑥 = 𝐿
Limits!