Calculus Revision Summary
1. The formula for the average gradient between two points on a curve is the same as the gradient of a straight line
y2 y1 f ( x h) f ( x)
Average gradient
x2 x1 h
2. Finding the derivative ( or gradient at a point) from First Principles:(formula is on the formula sheet)
f ( x+ h)−f ( x )
' lim
f (x) = h→0 h
NB
Watch your notation ; put f (x) in a bracket ; all the x’s should cancel ; take out h as a common factor to cancel the h in
the denomination; find the limit by substitution in h= 0 ; check your answer by using the rule
dy
f '( x) or or y ' or D x
3. Differentiation using the Rule : Notations dx
If y x n then y ' nx n 1 and If y k (a constant) y ' 0
In Grade 12 we learn how to differentiate expressions that are the sum or difference of terms
because the derivative of a sum ( or difference) is the sum ( or difference) of the derivatives
NB : Prepare f (x) for differentiation
1
1) multiply out products 2) get rid of root signs x x 2
1 x2 1 1 1 1 3 x 2 x 2 ( x 2)( x 1)
x 1 or bydivision x x or by factorisation x 2
3) get rid of fractions x 2 x3 2 2 x 1 x 1
4. Equation of the tangent at a point on a curve
The point or just the x value will be given
1) find the y value at the given x value - this is ( x; f ( x ))
2) find the gradient of the tangent m f '( x )
y mx c or y y1 m( x x1 )
3) Substitute ( x; f ( x)) into the equation to find c
m f '( x)
Any mention of a tangent means tangent
Remember parallel lines have equal gradients
5. Curve Sketching
3 2
Steps to sketching the curve y ax bx cx d
1. Determine the shape ( of a > 0 right arm goes up)
2. Find the y –intercept ( x = 0 ; y = d)
3. Find the x – intercepts :
Factorize the cubic equation (use the factor theorem and division by inspection to factorize the cubic
equation – if necessary)
The Factor theorem says if f (a) = 0 then (x−a) is a factor
4. Calculate the Turning Points ( Stationary Points / Max and Min ) by
' '
a) Finding f ( x ) b) putting f ( x )=0
c) Find the x-coordinates and calculate the corresponding y – coordinates by substituting the x coordinates
into the original equation i.e into f ( x )
1. The formula for the average gradient between two points on a curve is the same as the gradient of a straight line
y2 y1 f ( x h) f ( x)
Average gradient
x2 x1 h
2. Finding the derivative ( or gradient at a point) from First Principles:(formula is on the formula sheet)
f ( x+ h)−f ( x )
' lim
f (x) = h→0 h
NB
Watch your notation ; put f (x) in a bracket ; all the x’s should cancel ; take out h as a common factor to cancel the h in
the denomination; find the limit by substitution in h= 0 ; check your answer by using the rule
dy
f '( x) or or y ' or D x
3. Differentiation using the Rule : Notations dx
If y x n then y ' nx n 1 and If y k (a constant) y ' 0
In Grade 12 we learn how to differentiate expressions that are the sum or difference of terms
because the derivative of a sum ( or difference) is the sum ( or difference) of the derivatives
NB : Prepare f (x) for differentiation
1
1) multiply out products 2) get rid of root signs x x 2
1 x2 1 1 1 1 3 x 2 x 2 ( x 2)( x 1)
x 1 or bydivision x x or by factorisation x 2
3) get rid of fractions x 2 x3 2 2 x 1 x 1
4. Equation of the tangent at a point on a curve
The point or just the x value will be given
1) find the y value at the given x value - this is ( x; f ( x ))
2) find the gradient of the tangent m f '( x )
y mx c or y y1 m( x x1 )
3) Substitute ( x; f ( x)) into the equation to find c
m f '( x)
Any mention of a tangent means tangent
Remember parallel lines have equal gradients
5. Curve Sketching
3 2
Steps to sketching the curve y ax bx cx d
1. Determine the shape ( of a > 0 right arm goes up)
2. Find the y –intercept ( x = 0 ; y = d)
3. Find the x – intercepts :
Factorize the cubic equation (use the factor theorem and division by inspection to factorize the cubic
equation – if necessary)
The Factor theorem says if f (a) = 0 then (x−a) is a factor
4. Calculate the Turning Points ( Stationary Points / Max and Min ) by
' '
a) Finding f ( x ) b) putting f ( x )=0
c) Find the x-coordinates and calculate the corresponding y – coordinates by substituting the x coordinates
into the original equation i.e into f ( x )
