Unit 1 Lesson 2 Slopes of Tangents and Limits
Limits are used in mathematics to describe the value (output) that a function or
sequence approaches as the input approaches some value.
What is the limit of the sequence
What is the limit of
?
as x approaches 3?
Limits can be explored around a single input value or at ±∞ .
Limits
may
or
may not exist.
You have seen limits in advanced functions when exploring behaviour around
vertical asymptotes and end behaviours.
end behaviours
vertical asymptotes
1
Unit 1 Lesson 2 Slopes of Tangents and Limits
A secant (from Latin secare – to cut ) to a curve is a line which joins two points on the
curve.
A tangent (from Latin tangentem – to touch) to a curve is a line which touches the curve
at one point but does not intersect the curve at that point. The tangent to the curve will
be either above/below the curve on both sides of this point. The only exception occurs
when a tangent is drawn at a point of inflection.
Which lines
are tangents?
Which lines
are secants?
A tangent line will not exist at a point if the
tangent slope is dependent on whether you
approach the point from the left or the right.
Are there points on the curves for which the
tangent does not exist?
2
Unit 1 Lesson 2 Slopes of Tangents and Limits
Which line best
represents the
slope of the
tangent at (-2,6)?
Explain.
How could you create the best approximation for the slope of the tangent?
Given only one point, describe how it would be possible to determine the
slope of the tangent at that point.
3
Unit 1 Lesson 2 Slopes of Tangents and Limits
The defining equation of
a curve can be used to
determine an exact
measure of the slope of
the tangent at any point.
P(a,f(a)) is a fixed point on the curve y=f(x).
Q is another point on curve f(x) which is a horizontal distance of h units from
point P. Q has coordinates Q(a+h,f(a+h)).
The slope of the secant line PQ is given by:
a
The simplified version is known as the
Difference Quotient.
To find the slope of the tangent to f(x) at point P then find the limiting
slope of the secants PQ as Q approaches P, ie the horizontal difference
between points P and Q approaches zero.
Hence the slope of tangent, if it exists, is given by:
4
Unit 1 Lesson 2 Slopes of Tangents and Limits
2
1. Given f(x) = -2x + 7
a) Find the difference quotient needed to determine the slope of the tangent
at (-2,-1).
b) Is it possible to determine a difference quotient to find the slope of the
tangent to any point on the curve?
2. With respect to geometric representations (ie secants and tangents), what do
the following indicate about the graph of f(x)?
a)
b)
5
Unit 1 Lesson 2 Slopes of Tangents and Limits
3. Given
then
a) What is the equation of the function f(x)?
b) Will the difference quotient give the slope of the
tangent at x = 5? Explain.
2
4. Given f(x) = x - 3x + 4 then find the equation of the tangent to the curve at
the point (1,2).
6
Unit 1 Lesson 2 Slopes of Tangents and Limits
Thinking
2
Find the coordinates of the point on the curve f(x) = 3x - 4x where the tangent is
parallel to the line y = 8x.
7