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The curve is and point lies on the curve is .

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(a)

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Find the slope of the secant line .

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The point is .

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(i)

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If then the point is .

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

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Using slope of the two points is .

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(ii)

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If then the point is .

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

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Using slope of the two points is .

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(iii)

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If then the point is .

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

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Using slope of the two points is .

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(iv)

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If then the point is .

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

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Using slope of the two points is .

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(v)

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If then the point is .

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

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Using slope of the two points is .

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(vi)

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If then the point is .

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

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Using slope of the two points is .

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(vii)

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If then the point is .

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

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Using slope of the two points is .

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(viii)

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If then the point is .

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

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Using slope of the two points is .

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(b)

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Find the slope of the tangent line at .

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

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So near to the points are and .

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The slope of the secant line at is .

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The slope of the secant line at is .

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The average of the above slopes is

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The slope of the tangent line is .

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(c)

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Find the equation of the tangent line at .

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The slope of the tangent line is .

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

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Substitute and in the point-slope form.

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

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(d)

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The curve is .

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

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Consider any two points from to find the two secant lines.

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The slope of the secant line at is .

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The slope is and a point is .

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The equation of the first secant line is

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The slope of the secant line at is .

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The slope is and a point is .

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The equation of the second secant line is

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Graph :

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(1) Draw the coordinate plane.

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(2) Draw the curve .

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(3) Draw the tangent line at is

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(4) Draw the first secant line at is .

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(5) Draw the second secant line at is .

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(a)

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(i) , (ii) , (iii) , (iv) ,

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(v) , (vi) , (vii)  and (viii) .

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(b) The slope of the tangent line is .

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

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(d) Graph of the curve, tangent line and two secant lines is

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