Basic mathematics differentiation application

Basic mathematics differentiation application

• 1.
  Worked examples andexercises are in the text DIFFERENTIATION APPLICATIONS
• 2.
  Worked examples andexercises are in the text Tangents and normals to a curve at a given point Tangent 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 tangent can then be found from the equation: 1 1at ( , ) dy x y dx 1 1( ) where dy y y m x x m dx    
• 3.
  Worked examples andexercises are in the text Tangents and normals to a curve at a given point Example
• 4.
  Worked examples andexercises 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 1at ( , ) dy x y dx 1 1 1 ( ) where / y y m x x m dy dx     
• 5.
  Worked examples andexercises are in the text Tangents and normals to a curve at a given point Example Found the normal of the last exercise!
• 6.
  Worked examples andexercises are in the text Tangents and normals to a curve at a given point Exercise
• 7.
  Worked examples andexercises are in the text Try!
• 8.
  Worked examples andexercises 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
• 9.
  Worked examples andexercises are in the text The second derivative Example
• 10.
  Worked examples andexercises are in the text L’H 𝒐pital’s Rule for Forms of Type 𝟎 𝟎 and ∞ ∞ Example Both numerator and denominator have limit 0. Hence Try! , ,
• 11.
  Worked examples andexercises 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 = dTR dQ = -2Q + 15 -2Q + 15 = 0  Q = 7.5 Q = 7.5  TR = -(7.5)2 + 15.(7.5) = 56,25
• 12.
  Worked examples andexercises 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 = dTR dQ = -1.5Q + 13.4 MC = dTC dQ = 0.4Q2 + 4Q + 14 Profit  MR = MC
• 13.
  Worked examples andexercises 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±2.3 0.8  Q1 = 6 , Q2 = 0.25
• 14.
  Worked examples andexercises are in the text Inflection Points As 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 = 𝑥 1 3 + 2
• 15.
  Worked examples andexercises are in the text Try! Find the stationary points and points of inflexion on the following curve