MTH101 Final Term Solved Subjective Lectures 23-45

MTH101 Calculus And Analytical Geometry Lecture Wise · Questions and Answers For Final Term Exam Preparation · Lecture No 23 to 45 · Complete and Important Question and answer

Q1. What is the difference between definite integral and an indefinite integral?

If the function f is continuous on [a, b], and can assume both positive and negative values, the definite integral b

a f x dx

is net signed area between y = f(x) and the interval [a, b]. The numbers a and b are called the lower and upper limits of integration.

Indefinite integral is the set of functions F(x) + C, where C is constant of integration and F(x) is the integral of given function f where  d F x f x dx

Q2. What is meant by net signed area (the term used in defining definite integral)?

An indefinite integral is of the form f x dx F x C Suppose that a function f has a smooth curve in an interval [a, b] and can lie both above and below xaxis then ‘area under the curve and above x-axis’ minus ‘area above the curve below x-axis’ is termed as net signed area under the curve of f on the interval [a, b]. It is named "signed" area because the area above the x -axis counts as positive and the area below the x -axis counts as signed area can be positive, negative or zero; it is positive when there is more area above than below, negative when there is more area below than above, and zero when the area above and below are equal. For example, in the figure

Net signed area under the curve of f on the interval [a, b]

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= [Area above] – [Area below]

= (A + C) – B

Q3. Why we add C with the anti-derivative for evaluating an indefinite integral?

Answer: If a function f(x) is defined on an interval and F(x) is an antiderivative of f(x), then the indefinite integral is set of all anti-derivatives of f(x), that is, the functions F(x) + C, where C is an arbitrary constant.

As we know, the derivative of any constant function is zero. Once one has found one antiderivative F(x), adding or subtracting a constant C will give us another antiderivative, because (F(x) + C) ' = F ' (x) + C ' = F ' (x) . The constant is a way of expressing that every function has an infinite number of different anti-derivatives.

For example, suppose one wants to find anti-derivatives of cos(x). One such anti-derivative is sin(x). Another one is sin(x) + 1. A third is sin(x) − π. Each of these has derivative cos(x), so they are all anti-derivatives of cos(x).

It turns out that adding and subtracting constants is the only flexibility we have in finding different anti-derivatives of the same function. That is, all anti-derivatives are the same up to a constant. To express this fact for cos(x), we write:

∫ cos (x) dx = sin(x) + C

Q6. State First Fundamental Theorem of Calculus?

The First Fundamental Theorem of Calculus states that:

If f is continuous on the closed interval [a, b] and F is the anti-derivative of f on [ , ] a b , then

b

a f x dx F b F a

Q1. # Why we use the second fundamental theorem of calculus?

Answer: The first fundamental theorem of calculus is used to evaluate the definite integral of a continuous function if we can find an antiderivative for that function, but it does not point out the question of which functions actually have antiderivatives; then at this stage the second fundamental theorem of calculus is used.

The second fundamental theorem of calculus states that the derivative of a definite integral with respect to its upper endpoint is its integrand; it allows one to compute the definite integral of a function by using any one of its infinitely many antiderivatives. This part of the theorem has invaluable practical applications, because it markedly simplifies the computation of definite integral.

Q2. # Why the function defined by

x a F x f t dt

is not an arbitrary antiderivatives of f ; if

x a

Answer: The function defined by x a F x f t dt is not an arbitrary antiderivative of f ; it is

the specific antiderivative whose value at x a is zero because a a F a f t dt

Q1. What is the difference between local maximum and relative maximum value of a function?

Answer: Local maximum and relative maximum are synonyms, that is, different words for same concept. Similarly local minimum is also named as relative minimum.

Q2. What is the difference between critical point and stationary point of the function?

Answer: The term "critical point" is often confused with "stationary point".

Stationary point x0 is the point, at which the derivative of a function f(x) vanishes, f ’ ( x0) = 0 That is, a point where the function "stops" increasing or decreasing (hence the name).

Critical point x0 of a function y = f(x) is the point at which f ’ ( x0) = 0