Respuesta :

Space

Answer:

[tex]\displaystyle f''(1) = \frac{7}{\sqrt{43}}[/tex]

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

  1. Brackets
  2. Parenthesis
  3. Exponents
  4. Multiplication
  5. Division
  6. Addition
  7. Subtraction
  • Left to Right

Algebra I

Point-Slope Form: y - y₁ = m(x - x₁)  

  • x₁ - x coordinate
  • y₁ - y coordinate
  • m - slope

Calculus

Derivatives

Derivative Notation

Taking Derivatives with respect to x

Basic Power Rule:

  • f(x) = cxⁿ
  • f’(x) = c·nxⁿ⁻¹

Chain Rule: [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]

Step-by-step explanation:

Step 1: Define

Function: y = f(x), twice differentiable.

[tex]\displaystyle f(1) = 3\\\frac{dy}{dx} = (4y^2 + 7x^2)^{\frac{1}{2}}[/tex]

Step 2: Differentiate

Remember we are taking the derivative with respect to x.

  1. Chain Rule [Basic Power Rule]:                                                                     [tex]\displaystyle \frac{d^2y}{dx^2} = \frac{1}{2}(4y^2 + 7x^2)^{\frac{1}{2} - 1} \cdot \frac{dy}{dx} (4y^2 + 7x^2)[/tex]
  2. [2nd Derivative] Simplify:                                                                               [tex]\displaystyle \frac{d^2y}{dx^2} = \frac{1}{2}(4y^2 + 7x^2)^{-\frac{1}{2}} \cdot \frac{dy}{dx} (4y^2 + 7x^2)[/tex]
  3. [2nd Derivative - Chain Rule] Basic Power Rule:                                         [tex]\displaystyle \frac{d^2y}{dx^2} = \frac{1}{2}(4y^2 + 7x^2)^{-\frac{1}{2}} \cdot (0 + 2 \cdot 7x^{2 - 1})[/tex]
  4. [2nd Derivative] Simplify:                                                                               [tex]\displaystyle \frac{d^2y}{dx^2} = \frac{1}{2}(4y^2 + 7x^2)^{-\frac{1}{2}} \cdot 2 \cdot 7x[/tex]
  5. [2nd Derivative] Simplify:                                                                               [tex]\displaystyle \frac{d^2y}{dx^2} = \frac{7x}{\sqrt{7x^2 + 4y^2}}[/tex]

Step 3: Evaluate

We are given x = 1 and f(1) = 3. This will tell us the instantaneous slope.

  1. Substitute [2nd Deriv]:                     [tex]\displaystyle f''(1) = \frac{7(1)}{\sqrt{7(1)^2 + 4(3)^2}}[/tex]
  2. [√Radical] Exponents:                     [tex]\displaystyle f''(1) = \frac{7(1)}{\sqrt{7(1) + 4(9)}}[/tex]
  3. [Fraction] Multiply:                           [tex]\displaystyle f''(1) = \frac{7}{\sqrt{7 + 36}}[/tex]
  4. [√Radical] Add:                               [tex]\displaystyle f''(1) = \frac{7}{\sqrt{43}}[/tex]

This tells us that the rate of change of the slope of the tangent line is [tex]\displaystyle \frac{7}{\sqrt{43}}[/tex].

We can also write an equation for the instantaneous slope:

[Equation] [tex]\displaystyle y - 3 = \frac{7}{\sqrt{43}}(x - 1)[/tex]

Otras preguntas