Question

Find the third degree Taylor polynomial T₃(x) for f(x) = 2x³ - 5x² + x - 7 centered at a = 1.

Answer 1

To find the third degree Taylor polynomial for f(x) = 2x³ - 5x² + x - 7 centered at a = 1, we compute the function's first three derivatives, evaluate them at x = 1, and use the results to construct T3(x) = -9 - 3(x - 1) + 1(x - 1)² + 2(x - 1)³.

Step-by-step explanation:

To find the third degree Taylor polynomial T3(x) for the given function f(x) = 2x³ - 5x² + x - 7 centered at a = 1, we'll first calculate the derivatives of f(x) up to the third order and evaluate them at x = 1.

• f'(x) = 6x² - 10x + 1
• f''(x) = 12x - 10
• f'''(x) = 12

Evaluating these derivatives at x = 1 gives us:

• f'(1) = 6(1)² - 10(1) + 1 = -3
• f''(1) = 12(1) - 10 = 2
• f'''(1) = 12

Now we use these values to determine the polynomial coefficients:

• The constant term is f(1) = 2(1)³ - 5(1)² + 1 - 7 = -9.
• The linear coefficient is f'(1) / 1! = -3.
• The quadratic coefficient is f''(1) / 2! = 1.
• The cubic coefficient is f'''(1) / 3! = 2.

Thus, the third degree Taylor polynomial centered at a = 1 for the function is:

T3(x) = -9 - 3(x - 1) + 1(x - 1)² + 2(x - 1)³