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Mr. J Gallagher
Differentiation Worksheet
First Principles
Differentiate the following from first principles:
a) x2
b) 2x2 – 3x – 6
c) 3x2 + 7x
d) x2 + 6x
Tangent to a curve
Given Point on Tangent:
a) Find the equation of the tangent to the curve f (x) = 2x2 – 4x – 5 at the point (3,1).
b) Find the equation of the tangent to the curve f (x) = x2 – 6x at the point where x = 2.
c) Find the equation of the tangent to the curve f (x) = 2x2 – 4x – 5 at the point (3,1).
d) Find the equation of the tangent to the curve f (x) = 5 x2
1+ x2 at the point
(2,4).
Given Slope:
a) Find the coordinates of the point on the curve y = x2 + 3x – 1 at which the slope of the tangent to the curve is 5.
b) Find the coordinates of the point on the curve y = x2 + 4x + 6 at which the slope of the tangent to the curve is –2.
c) Let f (x) = 2 xx+2 , x ≠2, x ∈ R. Find the coordinates of the points at which
the slope of the tangent to the curve is 14 .
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Mr. J Gallagher
Given Point and Slope:
a) Find the value of k if the slope of the tangent to the curve y = x2 + kx is 3 at the point where x = -1.
b) A curve is defined by the equation y = ax2 + b, where a, b are constants. If the slope of the curve at the point (2, -2) is 3. Find the values of a and b.
c) Given that the curve with equation y = ax2 + bx + 5 has slope 4 at the point (5,0). Find the value of the constants a and b.
Maximum / Minimum
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.
State whether the functions are Injective, Surjective or Bijective. Give a reason for your answer.
a) f (x) = x3 – 9x2 + 24x – 20.
b) f (x) = x3 – 6x2 + 9x + 2
c) f (x) = x3 + 3x2 + 1
d)
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Real Life Examples
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Question 9
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Question 13
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Question 16
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Question 17
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