Everything Maths Grade 11

CHAPTER 14. GRADIENT AT A POINT 14.2

Example 1: Average Gradient

QUESTION

Find the average gradient between two points P(a; g(a)) and Q(a + h; g(a + h)) on a curve

g(x) = x^2. Then find the average gradient between P(2; g(2)) and Q(4; g(4)). Finally, explain

what happens to the average gradient if P moves closer to Q.

SOLUTION

Step 1 : Label x points

x 1 = a

x 2 = a + h

Step 2 : Determine y coordinates

Using the function g(x) = x^2 , we can determine:

y 1 = g(a) = a^2

y 2 = g(a + h)

= (a + h)^2

= a^2 + 2ah + h^2

Step 3 : Calculate average gradient

y 2 − y 1

x 2 − x 1

=

(a^2 + 2ah + h^2 )− (a^2 )

(a + h)− (a)

=

a^2 + 2ah + h^2 − a^2

a + h− a

=

2 ah + h^2

h

=

h(2a + h)

h

= 2a + h (14.1)

The average gradient between P(a; g(a)) and Q(a + h; g(a + h)) on the

curve g(x) = x^2 is 2 a + h.

Step 4 : Calculate the average gradient between P(2; g(2)) and Q(4; g(4))

We can use the result in(14.1), but we have to determine what a and h are. We

do this by looking at thedefinitions of P and Q. The x-coordinate of P is a and

the x-coordinate of Q is a + h therefore if we assumethat a = 2 and a + h = 4,

then h = 2.

Then the average gradient is:

2 a + h = 2(2) + (2) = 6

Step 5 : When P moves closer to Q