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Determine the value of k

Determine the value of k-example-1
User Scar
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2 Answers

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Answer:

k=6

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.

Determine the value of k-example-1
User Marcel Bochtler
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7.1k points
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Answer:


\displaystyle k = 6

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

Explanation:

Step 1: Define

Identify

Continuous at x = 2


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

Step 2: Solve for k

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

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

Unit: Limits - Continuity

Book: College Calculus 10e

User AJak
by
7.9k points

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