P1^

6

Reversing differentiation

Integration

Many small make a great.

Chaucer

●?^ In^ what^ way^ can^ you^ say^

that these four curves are

all parallel to each other? 

Reversing differentiation

In some situations you know the gradient function, d

d

y

x

, and want to find the

function itself, y. For example, you might know that d

d

y

x

=   2 x and want to find y.

You know from the previous chapter that if y = x^2 then d

d

y

x

=   2 x, but

y = x^2 + 1, y = x^2 −  2    and many other functions also give d

d

y

x

=   2 x.

Suppose that f(x) is a function with f′(x) =    2 x. Let g(x) = f(x) − x^2.

Then g′(x) = f′(x) −    2 x =   2 x −   2 x =   0    for all x. So the graph of y = g(x) has zero

gradient everywhere, i.e. the graph is a horizontal straight line.

Thus g(x) = c (a constant). Therefore f(x) = x^2 + c.

All that you can say at this point is that if d

d

y

x

=   2 x then y = x^2 + c where c is

described as an arbitrary constant. An arbitrary constant may take any value.

O

y = x^3 + 4

y = x^3 + 7

y = x^3

y = x^3 – 2

x

y

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