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BASIC CALCULUS
MODULE 1
THE LIMIT OF A FUNCTION
Definition of Limit of a Function โ Let f be a function at every number in some open interval containing c, except possibly at the number c itself. If the value of f is arbitrarily close to the number L for all the values of x sufficiently close to c, then the limit of f(x) as x approaches c is L. In symbols,
Limit and Function Value
Existence of Limit โ The limit of a function as x โ c exists if โ f(c) is defined ; or โ if f(c) is not defined , then f must approach the same value as x moves closer to c from both directions. By Marcverick Raboy - RCC BOARD MEMBERS
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MODULE 2
ILLUSTRATION OF LIMIT LAWS
MODULE 3
LIMITS OF POLYNOMIAL, RATIONAL, &
RADICAL FUNCTIONS
By Marcverick Raboy - RCC BOARD MEMBERS
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Evaluating Limits of Trigonometric
Functions
MODULE 5
LIMIT OF SOME TRANSCENDENTAL
FUNCTIONS
MODULE 6
CONTINUITY OF FUNCTIONS AT A POINT
โ Let us use the graph to check if the function is continuous at x = 1. Note that one is able to trace the graph from the left side of the number x = 1 going to the right side of x = 1, without lifting oneโs pen. Hence, we can say that the function is continuous at x = 1. By Marcverick Raboy - RCC BOARD MEMBERS
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MODULE 7
CONTINUITY OF A FUNCTION ON AN
INTERVAL
One-Sided Continuity
Continuity of Polynomial, Absolute Value,
Rational and Square Root Functions
โ (a) Polynomial functions are continuous everywhere. โ (b) The absolute value function f(x) = |x| is continuous everywhere. โ (c) Rational functions are continuous on their respective domains. โ (d) The square root function f(x) = โx is continuous on [0, โ). MODULE 8
PROBLEM INVOLVING CONTINUITY
Intermediate Value Theorem (IVT)
โ If a function f is continuous on the closed interval [a, b] and if f(a) โ f(b),then for any number k between f(a) and f(b), there exists at least one number c between a and b such that f(c) = k. By Marcverick Raboy - RCC BOARD MEMBERS
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MODULE 10
INTRODUCTION TO DERIVATIVES (2)
โ The derivative of a function f (x) denoted f '(x) at any X in the domain of the given function is defined as: NOTE : The process of solving the derivative is called differentiation. โ The techniques in differentiating functions is the same with evaluating limits and in indeterminate form 0/0, factoring and rationalization process could be utilized. MODULE 11
INTRODUCTION TO DERIVATIVES (3)
MODULE 12
DERIVATIVE OF ALGEBRAIC,
EXPONENTIAL AND LOGARITHMIC
FUNCTION
By Marcverick Raboy - RCC BOARD MEMBERS
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DIFFERENTIATION RULES
By Marcverick Raboy - RCC BOARD MEMBERS
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MODULE 15
SOLVING OPTIMIZATION PROBLEMS
โ Optimize means that we want to maximize or minimize the quantity. STEPS IN SOLVING OPTIMIZATION PROBLEMS
- Read and understand the problem.
- Draw a diagram , if applicable.
- Write down formulas and given information.
- Write a function for what is to be optimized.
- Set the derivative of the function equal to zero and solve. MODULE 16
THE CHAIN RULE OF DIFFERENTIATION
โ If y is a function of u, and u is a function of x, then: By Marcverick Raboy - RCC BOARD MEMBERS
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โ The chain rule is probably the trickiest among any other rules in differentiation. MODULE 17
IMPLICIT DIFFERENTIATION
โ Not all functions can be easily written in a form where the independent variable is completely isolated from the dependent variable, and for some relations it is simply not possible. Functions and relations of these types are called implicit. MODULE 18
RELATED RATES
โ This is essentially the study of how two or more quantities that change with time are connected and can be linked with an equation in which the relation of their rates of change may be found by differentiating both sides of the equation.
STEPS IN SOLVING PROBLEMS INVOLVING
RELATED RATES
- Read and understand the problem.
- Make a carefully labelled diagram.
- Write down and label constants, variables, rates and what is being sought.
- Write a function that relates the variables.
- Differentiate all terms with respect to time.
- Substitute known quantities. By Marcverick Raboy - RCC BOARD MEMBERS
