Question

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.​

Answer 1

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

Answer 2

Answer:

a = -1

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

Calculus

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:

Step 1: Define

y = 2x³ + ax² - x + 3

y'(1) = 3

Step 2: Differentiate

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

Step 3: Solve for a

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 a term:                   -2 = 2a
6. Isolate a:                            -1 = a
7. Rewrite:                             a = -1