Ch. 3 - Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 3 - Derivatives

Problema 55c

Capitolo 3, Problema 55c

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.

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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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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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06:51

Derivatives Applied To Acceleration Example 2

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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Percorso guidato

5:25

Intro to Transformations

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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Determining Differentiability Graphically

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Quadratic approximations

[Technology Exercise] e. Find the quadratic approximation to h(x) = √(1 + x) at x = 0. Graph h and its quadratic approximation together. Comment on what you see.

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A weight is attached to a spring and reaches its equilibrium position (x = 0). It is then set in motion resulting in a displacement of x = 10 cos t, where x is measured in centimeters and t is measured in seconds. See the accompanying figure.

Find the spring’s displacement when t = 0, t = π/3, and t = 3π/4.

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When the length L of a clock pendulum is held constant by controlling its temperature, the pendulum’s period T depends on the acceleration of gravity g. The period will therefore vary slightly as the clock is moved from place to place on Earth’s surface, depending on the change in g. By keeping track of ΔT, we can estimate the variation in g from the equation T = 2π(L/g)¹/² that relates T, g, and L.

b. If g increases, will T increase or decrease? Will a pendulum clock speed up or slow down? Explain.

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Assume that a particle’s position on the x-axis is given by

x = 3 cos t + 4 sin t,

where x is measured in feet and t is measured in seconds.

a. Find the particle’s position when t = 0, t = π/2, and t = π.

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b. Find the spring’s velocity when t = 0, t = π/3, and t = 3π/4.