MATH 101 Assignment 3 Question 3

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Antiderivative
If we know the derivative of a function, the process that we try to find its origin function is the antiderivative.

However, the derivative of a constant is 0, we cannot determine which constant is in the origin function, so we can just find a set of functions which all have a derivative like this.

For instance, if f(t)=t is the derivative of F(x)

We need to figure out which function has a derivative equal to t.
By doing the opposite operation of differentiation, it is easy to find out one of the origin function F(t)= ½ t^2
As we know, the derivative of a constant is 0, so we cannot determine what the certain original function is. In fact, all the function F(t)=½ t^2 +C (C can be any number) have a derivative equal to f(t)

Integration

Integration can be classified into definite integration and indefinite integration. Indefinite integration exactly has the same meaning of antiderivative. The indefinite integration of a function f, which is also known as the antiderivative of a function f, is a function F such that F′=f, while the definite integral is just a number. The definite integration of a function is equal to the area under the function on a certain domain.

As the picture shows above, the grey part is the indefinite integral of the function f(t).
If we replace variate t with a certain number, it becomes the definite integration of the function f(t).

MATH 100 ASSIGNMENT 7 Question 3b)

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I have solved Question 3 on the MATH 100 V01 Midterm exam.

a) I have memorized the definition of continuity. When the limit of g(x) as x approaches a should reach a number of g(a), because it is getting extremely close to a, so that we say that g(x) is equal to g(a) when x goes to a.

b) Same concept is applied in this question as part a. Using the definition of continuity, in order to make f(w) to be continuous at g(a), the limit of f(w) as w goes to g(a) should go to f(g(a)).

c) Recall part (a) and (b). We can say that the limit of g(x) as x approaches a is also equal to f(g(a)) due to the inter relationship between the functions. Therefore, we can say that the limit of f(w) as w goes to a is equal to f(g(a)) which is equal to the limit of f(g(x)) as x approaches a.