basic mathematics differentiation application
TRANSCRIPT
Worked examples and exercises are in the text
DIFFERENTIATION APPLICATIONS
Worked examples and exercises are in the text
Tangents and normals to a curve at a given point
TangentThe gradient of a curve, y = f
(x), at a point P with
coordinates (x1, y1) is given by
the derivative of y (the gradient
of the tangent) at the point:
The equation of the tangent can
then be found from the
equation:
1 1 at ( , )dy
x ydx
1 1( ) where dy
y y m x x mdx
Worked examples and exercises are in the text
Tangents and normals to a curve at a given point
Example
Worked examples and exercises are in the text
Tangents and normals to a curve at a given point
Normal
The gradient of a curve, y = f (x), at a point P with
coordinates (x1, y1) is given by the derivative of y (the
gradient of the tangent) at the point:
The equation of the normal (perpendicular to the
tangent) can then be found from the equation:
1 1 at ( , )dy
x ydx
1 1
1( ) where
/y y m x x m
dy dx
Worked examples and exercises are in the text
Tangents and normals to a curve at a given pointExample
Found the normal of the last exercise!
Worked examples and exercises are in the text
Tangents and normals to a curve at a given point
Exercise
Worked examples and exercises are in the text
Try!
Worked examples and exercises are in the text
Maximum and Minimum Value
See the figure below
Points A, B and C are called stationary points on the graph.
From the first derivative curve, we see that for stationary
points
Worked examples and exercises are in the text
The second derivative
Example
Worked examples and exercises are in the text
L’H 𝒐pital’s Rule for Forms of Type 𝟎
𝟎and
∞
∞
Example
Both numerator and denominator have limit 0. Hence
Try! , ,
Worked examples and exercises are in the text
Total Revenue of an economic function
Exercise
*Diketahui fungsi total revenue TR = - Q2 + 15Q, pada
jumlah produk Q berapa akan mencapai revenue
maksimum ? Dan berapakah revenue maksimum ?
TRmax MR = 0 , MR = dTRdQ
= -2Q + 15
-2Q + 15 = 0 Q = 7.5
Q = 7.5 TR = -(7.5)2 + 15.(7.5) = 56,25
Worked examples and exercises are in the text
Profit of an economic function
Exercise
*Diketahui fungsi total revenue TR = -0.75Q2 + 13.4Q dan
TC = 2/15Q3 – 2Q2 + 14Q +8. Tentukan besarnya produksi
agar diperoleh laba !
MR = dTRdQ
= -1.5Q + 13.4
MC = dTCdQ
= 0.4Q2 + 4Q + 14
Profit MR = MC
Worked examples and exercises are in the text
MR = MC
-1.5Q + 13.4 = 0.4Q2 – 4Q + 14
0.4Q2 – 2.5Q + 0.6 = 0
Q = −b± b2−4a.c
2a=−(−2.5)± (−2.5)2−4(0.4).(0.6)
2.(0.4)
= 2.5±2.30.8
Q1 = 6 , Q2 = 0.25
Worked examples and exercises are in the text
Inflection PointsAs you might guess, points where 𝑓′′ 𝑥 = 0 𝑜𝑟 𝑤ℎ𝑒𝑟𝑒 𝑓′′ 𝑥 doesn’t exist are
candidates for points of inflection. We use the word candidate deliberately. Just as
a candidate for political office, may fail to be elected, so, for example, may a point
where 𝑓′′ 𝑥 = 0 fail to be a point of inflection. Consider 𝑓 𝑥 = 𝑥4, which has
the graph shown in figure below. It is true that 𝑓′′ 0 = 0 ; yet the origin isn’t a
point of inflection. Therefore, in searching for inflection points, we begin by
identifying those point where 𝑓′′ 𝑥 = 0 (and where 𝑓′′ 𝑥 does not exist). Then
we check to see if they really are inflection points.
Example
Find all the points of the inflection of 𝐹 𝑥 = 𝑥1
3 + 2
Worked examples and exercises are in the text
Try!
Find the stationary points and points of inflexion on
the following curve