Mr. J Gallagher Name: ______________________________

Transition Year CalculusSketching Graphs

Example 1:Sketch the function f(x) = – x2 + 4x + 5 in the domain – 2 ≤ x ≤ 6. Hence find where the curve crosses the x & y-axis. Locate the maximum / minimum point of the function.

Example 2:Sketch the function f(x) = – x3 – 2x2 + 5x + 6 in the domain – 4 ≤ x ≤ 3. Hence find where the curve crosses the x & y-axis.Locate the maximum / minimum point of the function.

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Mr. J Gallagher Name: ______________________________

Differentiation by Rule

If y = xn, then dydx = nxn – 1 for all n ∈ R.

If y = axn, then dydx = naxn – 1

Example 1:Find the derivative of the following:

a) y = 5x2 – 4x

b) y = 6x2 + 5x + 4

c) f(x) = x2 + 2x + 1x

d) f(x) = x3 – 8x + 2

e) f(x) = 2x3 + x2 – 1x2

***Multiply the coefficient by the ***power and reduce the power by 1.

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f) y = √ x + 1√x

Example 2:a) Find the derivative of f (x) = 6x3 – 3x2 + 4x + 2.

Hence find f ’(2) and interpret the result.

b) Find the derivative of f (x) = 2x3 + 3x2 – x + 15.Hence find f ’(-2) and interpret the result.

c) Find the slope and hence the equation of the tangent to the curve y = 6 + x – x2 at the point (2, 4).

d) Find the points on the curve y = x3 – 3x2 at which the slope of the tangent to the curve is 9.

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e) If f (x) = x3 + 2√ x , find f ‘(x) and hence evaluate f ‘(4).

f) Find the slope of the tangent to the curve y = x2 – 2x – 3 at the point (2, 3).

g) Find the points on the curve y = 2x2 – x – 4 at which the slope of the tangent to the curve is 3.

h) Show that the tangent to the curve y = x2 – 3x + 4 at the point

where x = 1.5 is parallel to the x-axis.

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Mr. J Gallagher Name: ______________________________

Second Derivative

Example 1:

Find d2 yd x2 of the following functions:

a) y = x3 + 2x2

b) y = x4 – 3x2 + 6

c) f (x) = 3x3 – 3x2 + 6x + 2

d) y = √ x

e) f (x) = 3x + 4x

f) Given that f (x) = 2x + 4√ x , show that f “(4) = −1

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Mr. J Gallagher Name: ______________________________

Maximum / MinimumExample 1:

a) Find the turning point of the function y = x2 – 4x + 9 and show that this is a local minimum.

b) Find the turning point of the function y = 4 – 8x – 2x2 and show that this is a local maximum.

c) Find the turning point of the function y = 3x2 – 6x + 4 and determine if it is a local minimum or local maximum.

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d) Find the maximum and minimum turning points of the curve f (x) = x3 – 9x2 + 15x + 2.

e) Find the maximum and minimum turning points of the curve f (x) = x3 – 9x2 + 24x + 20.

f) Find the maximum and minimum turning points of the curve f (x) = x3 – 6x2 + 9x + 2.

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Mr. J Gallagher Name: ______________________________

Example 2:

For each of the following functions:

Calculate the maximum and minimum values.

Find the point(s) where the curve crosses the y-axis.

Hence draw a sketch of the curve.

Give the range of values for which the curve is increasing / decreasing.

Calculate the point of inflection.

a) f (x) = x3 – 9x2 + 24x – 20.

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b) f (x) = x3 – 6x2 + 9x + 2

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c) f (x) = x3 + 3x2 + 1

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d)

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Real Life Examples

Question 1

Question 2

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Question 3

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Question 5

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Question 7

Question 8

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