Motion In A Straight Line
Concepts of Differentiation and Integration
Calculus
Calculus is basically a way of calculating rate of changes (similar to slopes, but called
derivatives in calculus), areas, volumes, and surface areas (for starters).
It’s easy to calculate these kinds of things with algebra and geometry if the shapes
you’re interested in are simple. For example, if you have a straight line you can
calculate the slope easily.
But if you want to know the slope at an arbitrary point (any random point) on the graph
of some function like x-squared or some other polynomial, then you would need to use
calculus. In this case, calculus gives you a way of “zooming in” on the point you’re
interested in to find the slope exactly at the point. This called a derivative.
If you have a cube or a sphere, you can calculate the volume and surface area easily. If
you have an odd shape, you need to use calculus. You use calculus to make a infinite
number or really small slices of the object you’re interested in, determine to sizes of the
slices, and then add all those sizes up. This process is called integration. It turns out
that integration is the reverse of derivation (finding a derivative).
In summary, calculus is a tool that lets you do calculation with complicated curves
shapes, etc. that you would normally not be able to do with just algebra and geometry.
Differentiation and Integration
Differentiation is the process of obtaining the derived function f′(x) from the function f(x),
where f′(x) is the derivative of f at x.
The derivatives of certain common functions are given in the Table of derivatives,
Table of derivatives :
f(x) f'(x)
xn nxn−-1
sin x cos x
cos x −-sin x
tan x sec2 x
cot x −-cosec2 x
, sec x sec x tan x
cosec x −-(cosec x )(cot x)
ln x 1/x
ex ex
Many other functions can be differentiated using the following rules of differentiation:
(i) If h(x) = k f(x) for all x, where k is a constant, then h′(x) = k f′(x).
(ii) If h(x) = f(x) + g(x) for all x, then h′ (x) = f′(x) + g′(x).
(iii) The product rule: If h(x) = f(x)g(x) for all x, then h′(x) = f(x)g′(x) + f′(x)+g′(x).
(iv) The reciprocal rule: If h(x) = 1/f(x) and f(x) ≠ 0 for all x, then
Integration is the process of finding an anti-derivative of a given function f. ‘Integrate f’
means ‘find an anti-derivative of f’. Such an anti-derivative may be called an indefinite
integral of f and be denoted by ∫f(x)dx.
The term ‘integration’ is also used for any method of evaluating a definite integral.
,The definite integral can be evaluated if an anti-derivative Φ of f can be found, because
then its value is Φ(b) − Φ(a). (This is provided that a and b both belong to an interval in
which f is continuous.)
However, for many functions f, there is no anti-derivative expressible in terms of
elementary functions, and other methods for evaluating the definite integral have to be
sought, one such being so-called numerical integration.
Example:
Differential and Integral Calculus
Differential calculus
Let x and y be two quantities interrelated in such a way that for each value of x there is
one and only one value of y.
The graph represents the y versus x curve. Any point in the graph gives an unique
values of x and y. Let us consider the point A on the graph. We shall increase x by a
small amount Δx, and the corresponding change in y be Δy.
, Thus, when x change by Δx, y change by Δy and the rate of change of y with respect
to x is equal to
In the triangle ABC, coordinate of A is (x, y); coordinate of B is (x + Δx, y + Δy)
The rate can be written as,
But this cannot be the precise definition of the rate because the rate also varies
between the point A and B. So, we must take very small change in x. That is Δx is
nearly equal to zero. As we make Δx smaller and smaller the slope tanθ of the line AB
approaches the slope of the tangent at A. This slope of the tangent at A gives the rate of
change of y with respect to x at A.
This rate is denoted by
and,
Question:
Find the slope of the curve y = 1 + x2 at x = 5.
Solution:
y = 1 + x2
We know slope is given by,
Concepts of Differentiation and Integration
Calculus
Calculus is basically a way of calculating rate of changes (similar to slopes, but called
derivatives in calculus), areas, volumes, and surface areas (for starters).
It’s easy to calculate these kinds of things with algebra and geometry if the shapes
you’re interested in are simple. For example, if you have a straight line you can
calculate the slope easily.
But if you want to know the slope at an arbitrary point (any random point) on the graph
of some function like x-squared or some other polynomial, then you would need to use
calculus. In this case, calculus gives you a way of “zooming in” on the point you’re
interested in to find the slope exactly at the point. This called a derivative.
If you have a cube or a sphere, you can calculate the volume and surface area easily. If
you have an odd shape, you need to use calculus. You use calculus to make a infinite
number or really small slices of the object you’re interested in, determine to sizes of the
slices, and then add all those sizes up. This process is called integration. It turns out
that integration is the reverse of derivation (finding a derivative).
In summary, calculus is a tool that lets you do calculation with complicated curves
shapes, etc. that you would normally not be able to do with just algebra and geometry.
Differentiation and Integration
Differentiation is the process of obtaining the derived function f′(x) from the function f(x),
where f′(x) is the derivative of f at x.
The derivatives of certain common functions are given in the Table of derivatives,
Table of derivatives :
f(x) f'(x)
xn nxn−-1
sin x cos x
cos x −-sin x
tan x sec2 x
cot x −-cosec2 x
, sec x sec x tan x
cosec x −-(cosec x )(cot x)
ln x 1/x
ex ex
Many other functions can be differentiated using the following rules of differentiation:
(i) If h(x) = k f(x) for all x, where k is a constant, then h′(x) = k f′(x).
(ii) If h(x) = f(x) + g(x) for all x, then h′ (x) = f′(x) + g′(x).
(iii) The product rule: If h(x) = f(x)g(x) for all x, then h′(x) = f(x)g′(x) + f′(x)+g′(x).
(iv) The reciprocal rule: If h(x) = 1/f(x) and f(x) ≠ 0 for all x, then
Integration is the process of finding an anti-derivative of a given function f. ‘Integrate f’
means ‘find an anti-derivative of f’. Such an anti-derivative may be called an indefinite
integral of f and be denoted by ∫f(x)dx.
The term ‘integration’ is also used for any method of evaluating a definite integral.
,The definite integral can be evaluated if an anti-derivative Φ of f can be found, because
then its value is Φ(b) − Φ(a). (This is provided that a and b both belong to an interval in
which f is continuous.)
However, for many functions f, there is no anti-derivative expressible in terms of
elementary functions, and other methods for evaluating the definite integral have to be
sought, one such being so-called numerical integration.
Example:
Differential and Integral Calculus
Differential calculus
Let x and y be two quantities interrelated in such a way that for each value of x there is
one and only one value of y.
The graph represents the y versus x curve. Any point in the graph gives an unique
values of x and y. Let us consider the point A on the graph. We shall increase x by a
small amount Δx, and the corresponding change in y be Δy.
, Thus, when x change by Δx, y change by Δy and the rate of change of y with respect
to x is equal to
In the triangle ABC, coordinate of A is (x, y); coordinate of B is (x + Δx, y + Δy)
The rate can be written as,
But this cannot be the precise definition of the rate because the rate also varies
between the point A and B. So, we must take very small change in x. That is Δx is
nearly equal to zero. As we make Δx smaller and smaller the slope tanθ of the line AB
approaches the slope of the tangent at A. This slope of the tangent at A gives the rate of
change of y with respect to x at A.
This rate is denoted by
and,
Question:
Find the slope of the curve y = 1 + x2 at x = 5.
Solution:
y = 1 + x2
We know slope is given by,
