The curve is and point lies on the curve is .

(a)

Find the slope of the secant line .

The point is .

(i)

If then the point is .

Use the points and to find the slope of the secant line .

Using slope of the two points is .

(ii)

If then the point is .

Use the points and to find the slope of the secant line .

Using slope of the two points is .

(iii)

If then the point is .

Use the points and to find the slope of the secant line .

Using slope of the two points is .

(iv)

If then the point is .

Use the points and to find the slope of the secant line .

Using slope of the two points is .

(v)

If then the point is .

Use the points and to find the slope of the secant line .

Using slope of the two points is .

(vi)

If then the point is .

Use the points and to find the slope of the secant line .

Using slope of the two points is .

(vii)

If then the point is .

Use the points and to find the slope of the secant line .

Using slope of the two points is .

(viii)

If then the point is .

Use the points and to find the slope of the secant line .

Using slope of the two points is .

(b)

Find the slope of the tangent line at .

Use the average of the slopes of the secant lines at .

So near to the points are and .

The slope of the secant line at is .

The slope of the secant line at is .

The average of the above slopes is

The slope of the tangent line is .

(c)

Find the equation of the tangent line at .

The slope of the tangent line is .

Use the point-slope form of a line equation is .

Substitute and in the point-slope form.

The equation of the tangent line to the curve at is .

(a)

(i) , (ii) , (iii) , (iv) ,

(v) , (vi) , (vii) and (viii) .

(b) The slope of the tangent line is .

(c) The equation of the tangent line to the curve at is .