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den301095 [7]
2 years ago
14

The gradient of the tangent of the curve y=2x^3 + ax^2 - x + 3 at the point x=1 is 3. Find the value of a.​

Mathematics
2 answers:
ArbitrLikvidat [17]2 years ago
5 0

Hello!

\large\boxed{a = -1}

y = 2x³ + ax² - x + 3

Find the derivative using the power rule:

y' = 6x² + 2ax - 1

Substitute in the given slope and x value to solve for a:

3 = 6(1)² + 2(1)a - 1

3 = 6 + 2a - 1

4 = 6 + 2a

-2 = 2a

a = -1

Vsevolod [243]2 years ago
3 0

Answer:

a = -1

General Formulas and Concepts:

<u>Pre-Algebra</u>

Order of Operations: BPEMDAS

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

Equality Properties

<u>Calculus</u>

The definition of a derivative is the slope of the tangent line.

The derivative of a constant is equal to 0.

Basic Power Rule:

  • f(x) = cxⁿ
  • f’(x) = c·nxⁿ⁻¹

Step-by-step explanation:

<u>Step 1: Define</u>

y = 2x³ + ax² - x + 3

y'(1) = 3

<u>Step 2: Differentiate</u>

  1. Basic Power Rule:                    y' = 3 · 2x³⁻¹ + 2 · ax²⁻¹ - 1 · x¹⁻¹ + 0
  2. Simplify:                                    y' = 6x² + 2ax - 1

<u>Step 3: Solve for </u><em><u>a</u></em>

  1. Substitute:                         3 = 6(1)² + 2a(1) - 1
  2. Exponents:                        3 = 6(1) + 2a(1) - 1
  3. Multiply:                             3 = 6 + 2a - 1
  4. Combine like terms:         3 = 5 + 2a
  5. Isolate <em>a</em> term:                   -2 = 2a
  6. Isolate <em>a</em>:                            -1 = a
  7. Rewrite:                             a = -1
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