Find the average gradient between the points (t;f(t)) and (t+h;f(t+h)) on the curve f(x)=x^2.

We'll put x1 = t and x2 = t+h and we'll calculate y1 and
y2, since we know that f(x) = y and f(x) = x^2

So, y1 =
f(x1) = x1^2= t^2

y2 = f(x2) = x2^2= (t+h)^2 = t^2 + 2th +
h^2

The average gradient
is:

(y2 - y1)/(x2 - x1) = (t^2 + 2th + h^2 - t^2)/(t + h -
t)

We'll eliminate like terms inside
brackets:

(y2 - y1)/(x2 - x1) = (2th +
h^2)/h

We'll factorize by
h:

(y2 - y1)/(x2 - x1) = h(2t +
h)/h

We'll simplify and we'll
get:

(y2 - y1)/(x2 - x1) = (2t +
h)

The average gradient between the given
points, on the curve f(x) = x^2 is (2t + h).