Textbook Question
Quadratic approximations
a. Let Q(x) = b₀ + b₁(x − a) + b₂(x − a)² be a quadratic approximation to f(x) at x = a with these properties:
i. Q(a) = f(a)
ii. Q′(a) = f′(a)
iii. Q″(a) = f″(a).
Determine the coefficients b₀, b₁, and b₂.
4. Applications of Derivatives
Differentials
1:49 minutesProblem 55c
Textbook Question
Quadratic approximations
[Technology Exercise] c. Graph f(x) = 1/(1 − x) and its quadratic approximation at x = 0. Then zoom in on the two graphs at the point (0,1). Comment on what you see.
Verified step by step guidance1
Identify the function f(x) = \(\frac{1}{1-x}\) and recognize that we need to find its quadratic approximation at x = 0.
Recall that the quadratic approximation of a function f(x) at a point a is given by the formula: \(f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2\).
Calculate the first derivative f'(x) of the function f(x) = \(\frac{1}{1-x}\) using the chain rule. The derivative is f'(x) = \(\frac{1}{(1-x)^2}\). Evaluate this at x = 0 to find f'(0).
Calculate the second derivative f''(x) of the function. Differentiate f'(x) = \(\frac{1}{(1-x)^2}\) again to get f''(x) = \(\frac{2}{(1-x)^3}\). Evaluate this at x = 0 to find f''(0).
Substitute f(0), f'(0), and f''(0) into the quadratic approximation formula to get the quadratic approximation of f(x) at x = 0. Graph both f(x) and its quadratic approximation, then zoom in on the point (0,1) to observe how closely the approximation matches the original function near this point.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Quadratic Approximation
Quadratic approximation is a method used to approximate a function near a point using a quadratic polynomial. It is derived from the Taylor series expansion, where the function is approximated by a polynomial of degree two. This approximation provides a more accurate representation of the function near the point of interest compared to a linear approximation.
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Taylor Series
The Taylor series is an infinite sum of terms calculated from the values of a function's derivatives at a single point. For a function f(x) at x = a, the Taylor series is f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + ..., which can be truncated to form polynomial approximations. The quadratic approximation is a specific case using terms up to the second derivative.
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Graphical Analysis
Graphical analysis involves visually examining the graphs of functions to understand their behavior and relationships. By graphing f(x) = 1/(1-x) and its quadratic approximation, one can observe how closely the approximation matches the original function near x = 0. Zooming in on the graphs at (0,1) helps to see the accuracy and limitations of the approximation in that region.
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