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

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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).**

We have the function f(x) = x^2.

The points given to us are (t , f(t)) and (t+h , f(t+h))

or (t , t^2) and ((t + h) , (t + h)^2)

The gradient between these points is

=>[ (t + h)^2 - t^2] / [ t + h - t]

=> (t + h - t)(t + h + t) / h

=> 2t + h

**The required gradient is 2t + h**

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