P1^
5
Exercise(^) 5A
129
EXERCISE 5A 1 Use the method in Example 5.1 to prove that the gradient of the curve y = x^2 at
the point (x, y) is equal to 2x.
2 Use the binomial theorem to expand (x + h)^4 and hence find the gradient of
the curve y = x^4 at the point (x, y).
3 Copy the table below, enter your answer to question 2 , and suggest how the
gradient pattern should continue when f(x) = x^5 , f(x) = x^6 and f(x) = xn (where
n is a positive whole number).
f(x) f'(x) (gradient at (x, y))
x^22 x
x^33 x^2
x^4
x^5
x^6
xn
4 Prove the result when f(x) = x^5.
Note
The result you should have obtained from question 3 is known as Wallis’s rule and
can be used as a formula.
●?^ How can you use the binomial theorem to prove this general result for integer values
of n?
An alternative notation
So far h has been used to denote the difference between the x co-ordinates of our
points P and Q, where Q is close to P.
h is sometimes replaced by δx. The Greek letter δ (delta) is shorthand for ‘a
small change in’ and so δx represents a small change in x and δy a corresponding
small change in y.
In figure 5.8 the gradient of the chord PQ is δ
δ
y
x
.
In the limit as δx → 0, δx and δy both become infinitesimally small and the value
obtained for δ
δy
x
approaches the gradient of the tangent at P.