DN1.3: GRADIENTS, TANG ENTS AND DERIVATIVES
DN1.3: GRADIENTS, TANGENTS AND DERIVATIVES
Gradient of a Curve
The gradient at a point on a curve is the gradient of the tangent to the curve at that point.
For the curve y = f(x) below, the gradient at point P is the gradient of the line AB. Special cases: horizontal and vertical lines
A line parallel to the x-axis with equation of the form y = k (k constant), has a gradient of zero.
As a line becomes closer to vertical it's gradient gets larger and larger. A line parallel to the y-axis with equation of the form x = c ( x constant) has a gradient which is undefined.
DN 1.3: Gradients, Tangents and Derivatives
Page 1 of 3
June 2012
This gradient cannot be calculated - only one point on the line is known. The gradient of the line PQ can be calculated and this can be used to approximate the gradient of AB.
f (x + h)- f (x)
The gradient of PQ = h
As the value of h decreases (i.e Q becomes closer to the point P), the approximation of the gradient is more accurate. The value of the gradient becomes most accurate as h approaches zero.
The gradient formula for the curve y = f(x) is defined as the derivative function
f (x + h) - f (x)
f (x) = lim
, h0
h0
h
The derivative function f (x) gives the slope of the tangent to the curve f (x) at any point x.
Example
3
1
-3
1. If the derivative function of f (x) = is f (x) = , find the slope of the tangent to the curve
x
x2
at x = 4
At x = 4,
1
f (x) =
f (4)
-3
=
4 2 -3
=
16
DN 1.3: Gradients, Tangents and Derivatives
Page 2 of 3
June 2012
Exercises
1. If the derivative function for this curve at a) x = 2 b) x = 0 c) x = -9
f (x) = x3 ? x is f (x) = 3x2 - 1, find the slope of the tangent to
2. If the derivative function of f (x) = sin(x) is f (x) = cos(x) find the gradient of y = sin(x) at
a) x = 0
b) x = 2
c) x = 3.5
(x + h)2 - x2
lim 3. Determine h0
h
and hence find the slope of the tangent to the curve y = x2 at
a) x = 2
b) x = 0
c) x = -9
Answers
1. a) 11 b) -1 c) 242
2. a) 1 b) 0 c) -0.94
3. lim (x + h)2 - x 2
h 0
h
a) 4
b) 0
c) -18
= 2x
DN 1.3: Gradients, Tangents and Derivatives
Page 3 of 3
June 2012
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