Differential Calculus - Applications of Derivative

Jumat, 26 Juni 2020

Applications of Derivative

 

Function: f, g, y

Position of an object: s

Velocity: v

Acceleration: w

Independent variable: x

Time: t

Natural number: n

 

 

• Velocity and Acceleration

   s = f (t) is the position of an object relative to a fixed coordinate system

   at a time t, v =   s’ = f’ (t) is the instantaneous velocity of the object, 

   w = v’ = s’’ = f’’ (t) is the instantaneous acceleration of the object.

 

• Tangent Line

   y – y₀ = f’ (x₀)(x – x₀)

 

Figure

 

 

• Normal Line

     

• Increasing and Decreasing Function.

   If  f’ (x₀) > 0, then f (x) is increasing at x₀. (Figure 1, x < x₁,  x₂ < x),

   If  f’ (x₀) < 0, then f (x) is decreasing at x₀. (Figure 1, x₁ < x₁ < x₂),

   If  f’ (x₀) does npt exist or is zero, then the test falis.

 

Figure

 

• Local extrema

   A function f (x) has a local maximum at x₁ if and only if there exists 

  some interval containing x₁ such that f (x₁) ≥ f (x) for all x in the interval 

  (Figure 2).

 

  A function f (x) has a local minimum at x₂ if and only if there exists some 

  interval containing x₂ such that f (x₂) ≤ f (x) for all x in the interval 

  (Figure 2).

  

• Critical Points

   A critical point on f (x) occurs at x₀ if and only if either f’ (x₀) is zero or 

   the derivation doesn’s exist.

 

• First Derivative Test for Local Extrema.

   If f (x) is increasing (f’ (x) > 0) for all x in some interval (a, x₁] and f (x) 

   is decreasing (f’ (x) < 0) for all x in some interval [x₁, b), then f (x) has a 

   local maximum at x₁ (Figure 2)

 

• If f (x) derivative (f’ (x) < 0) for all x in some increasing (a, x₂] and f (x) is increasing (f’ (x) > 0) for all x in some interval [x₂, b), then f (x) has a local minimum at x₂ (Figure 2)

 

• Second Derivative Test for Local Extrema.

   If f’ (x₁) = 0 and f’’ (x₁) < 0), then f (x) has a local maximum at x₁.  

   If f’ (x₂) = 0 and f’’ (x₂) > 0), then f (x) has a local minimum at x₂.  

   (Figure 2)

 

• Concavity.

   f (x) is concave upward at x₀ if and only if f’ (x) is increasing at x₀. 

  (Figure 2, x₃ < x)

   f (x) is concave downward at x₀ if and only if f’ (x) is decreasing at x₀. 

   (Figure 2, x₃ < x)

 

• Second Derivative Test for Concavity.

   If f’’ (x₀) > 0, then f (x) is concave upward at x₀.

   If f’’ (x₀) < 0, then f (x) is concave downward at x₀.

   If f’’ (x₀) does not exist or is zero, then the test fails.

 

• Inflection Points

   If f’ (x₃) exists and f’’ (x) changes sign at x = x₃ , then the point (x₃, f (x₃)) 

   is an inflection point of the graph of f (x). If f’’ (x₃) exists at the inflection 

   point, then f’’ (x₃) = 0. (Figure 2)

 

• L’Hopital’s Rule

     

Sumber

Labels: Mathematician

Thanks for reading Differential Calculus - Applications of Derivative . Please share...!