Respuesta :

Answer:

k=6

Step-by-step explanation:

For a piecewise function to be continuous, the left-side and right-side limits must be equal to each other. Therefore, the left-side limit of f(x) is 8 because the derivative of 2x^2 is 4x, and then directly substituting x=2 gives us 4(2)=8. Therefore, the right-side limit must also equal 8. Therefore, k must be 6 because 2+6=8.

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Answer:

[tex]\displaystyle k = 6[/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  

Equality Properties

  • Multiplication Property of Equality
  • Division Property of Equality
  • Addition Property of Equality
  • Subtraction Property of Equality

Algebra I

  • Functions
  • Function Notation

Algebra II

  • Piecewise Functions

Calculus

  • Limits
  • Continuity

Step-by-step explanation:

Step 1: Define

Identify

Continuous at x = 2

[tex]\displaystyle f(x) = \left \{ {{2x^2 \ if \ x < 2} \atop {x + k \ if \ x \geq 2}} \right.[/tex]

Step 2: Solve for k

  1. Definition of Continuity:                                                                                   [tex]\displaystyle \lim_{x \to 2^+} 2x^2 = \lim_{x \to 2^-} x + k[/tex]
  2. Evaluate limits:                                                                                                  [tex]\displaystyle 2(2)^2 = 2 + k[/tex]
  3. Evaluate exponents:                                                                                         [tex]\displaystyle 2(4) = 2 + k[/tex]
  4. Multiply:                                                                                                             [tex]\displaystyle 8 = 2 + k[/tex]
  5. [Subtraction Property of Equality] Subtract 2 on both sides:                        [tex]\displaystyle 6 = k[/tex]
  6. Rewrite:                                                                                                             [tex]\displaystyle k = 6[/tex]

Topic: AP Calculus AB/BC (Calculus I/II)

Unit: Limits - Continuity

Book: College Calculus 10e

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