Find gradients and tangents using limits

DIFFERENTIAL CALCULUS

Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points

If I have seen further…..

…. it is because I have stood on the shoulders of giants.

ISaac Newton to Robert Hooke in 1675

ISAAC NEWTON 1643 -1727



1



Introduction The two-point formula Gradients of secants

The concept of gradient is so important for a thorough
understanding of differential calculus.

The graphs of some linear functions are steep with a positive slope >

The graphs of some linear functions are less steep >

… and others have negative slopes >

Gradient is a measure of this steepness or slope.

It is defined as the ratio of the rise to the run.

The gradient of the green function is 2.

Check the gradient of the red function is 1/3

And the blue line has a gradient of-1

2




The given line passes through the points P and Q where:

P = ( 2, 3 )
P
Q = ( 1, 1 ) X

In the interval PQ:
rise
rise = 3 - 1

run = 2 - 1
Q
X
run

Generally, the straight line passing through the two points; ,

Has a gradient given by:

3




Using the formula for the gradient of a lineHere is the graph of the function

through two points, we have: And here is a secant PQ

where and .

4



Introduction Case study An algebraic approach Limiting process

The gradient of the graph of a linear function is easy to find; we can use the two-
point formula as shown in the previous section. But how can we find the gradient
at different points on a non-linear function, such as the one shown here?

Clearly the parabola gets steeper as the x-values increase…

… but how can we measure the actual gradient at any particular point on the curve?
Gradient of the function at the point ( 3, 9)

>
The gradient at a point P on the
DEFINITION

curve is defined as the gradient
of the tangent to the curve at Gradient of the function at the point ( 2, 4)
that point.

5




Our goal here is means finding the gradient of the tangent
By definition thisto find exactly the gradient of the function to the curve the point
at at that point…

As a first approximation, As a second approximation, For a third approximation
consider the secant AP consider the secant BP we will need to zoom in
and consider secant CP….
Note now how close the
tangent is to the curve

C

P

6




Our goal here is to find exactly the gradient of the function at the point

By definition this means finding the gradient of the tangent to the curve at that point…

As a first approximation, consider the secant

h

x+h

7




Using first principles find the derived function for

h

x+h

8




Using first principles find the derived function for Using first principles find the derived function for SUMMARY

8



Introduction Equation of a tangent Equation of a normal ACTIVITY 1

A tangent to a curve is a straight line touching the curve at a single point. A normal is a straight line, perpendicular to the tangent.

FIGURE 1 FIGURE 2 FACT SHEET

This diagram shows the TANGENT to the This diagram shows the NORMAL to the • A TANGENT touches a curve at a single
curve curve point
at the point (1, -2) at the point (1, -2)
• It’s gradient, , is given by the gradient or derived
function at the value

• Its equation is given by
NORMAL

• The NORMAL at a point is perpendicular to the
tangent at that point. (It’s at 90 degrees)

TANGENT • It’s gradient, , is found using the gradient of the
tangent and the fact that

• It’s equation is given by

9




Find the equation of the tangent to the curve at the point (2, 0)

METHOD

1 Differentiate to obtain the
gradient function

2 Find the gradient of the
function at x=2

3 Substitute the gradient,
m = 1 and the coordinates
of the point into the point/
gradient form of a straight
line.

10




Find the equation of the normal to the curve at the point (2, 0)

METHOD

1 Differentiate to obtain the
gradient function

function at x=2

normal at x=2

4 Substitute the gradient,
m = -1 and the
coordinates of the point into
the point/ gradient form of a
straight line.

11



Introduction Equation of a tangent Equation of a normal ACTIVITY


Home Limits Gradient First principles Tangents and Normals
2
Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points



MAXIMUM MINIMUM INFLEXION
STATIONARY POINT A STATIONARY POINT B POINT C A
LOCATED AT: LOCATED AT: LOCATED AT: C

B

FIRST DERIVATIVE IS ZERO FIRST DERIVATIVE IS ZERO

SECOND DERIVATIVE IS NEGATIVE SECOND DERIVATIVE IS POSITIVE SECOND DERIVATIVE IS ZERO



FIGURE 1 FIGURE 2 FACT SHEET

This diagram shows the STATIONARY POINT on This diagram shows the STATIONARY POINTS on • Stationary points lie on the graphs of functions where
the graph of the quadratic function the graph of the cubic function the gradient is zero. (The tangents to the curve
are horizontal at these points; the function is
neither increasing nor decreasing.)

• The stationary point in figure 1 is called a maxima.

• Figure 2 shows a function having both a maxima and
minima



Find the stationary point on the graph of the function

METHOD

1 Find the gradient
function
by differentiating

2 Find the x-value for
which the gradient is
zero
by solving the equation

3 Find the y-value of the
function at x = 1.5
by substitution into the
original function

4 Write down the
coordinates of the stationary
point



Find the stationary points on the function and determine their nature

METHOD

1 Differentiate the given
function to obtain the
gradient function

2 Find the x-values for
which the gradient
function is zero.

3 Substitute these x-values
into the original function to
determine the stationary
points.

4 Find the sign of the
second derivative to
determine the
nature of each stationary
point.

Find gradients and tangents using limits

Recomendados

Recomendados

Mais conteúdo relacionado

Mais procurados

Mais procurados (19)

Destaque

Destaque (20)

Semelhante a Find gradients and tangents using limits

Semelhante a Find gradients and tangents using limits (8)

Último

Último (20)

Find gradients and tangents using limits