Ch. 7 - Transcendental Functions

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

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problema 7.1.70c

Capitolo 7, Problema 7.1.70c

In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:

c. Find the equation for the tangent line to f at the specified point (x_0, f(x_0)).

70. y= x³/(x²+1), -1 ≤ x ≤ 1, x_0=1/2

Guida verificata passo dopo passo

First, identify the function given: \(f(x) = \frac{x^3}{x^2 + 1}\) and the point at which we want the tangent line: \(x_0 = \frac{1}{2}\).

Calculate the value of the function at \(x_0\) to find the point on the curve: compute \(f\left(\frac{1}{2}\right) = \frac{\left(\frac{1}{2}\right)^3}{\left(\frac{1}{2}\right)^2 + 1}\).

Find the derivative \(f'(x)\) using the quotient rule: if \(f(x) = \frac{u(x)}{v(x)}\), then \(f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}\). Here, \(u(x) = x^3\) and \(v(x) = x^2 + 1\).

Compute \(u'(x) = 3x^2\) and \(v'(x) = 2x\), then substitute into the quotient rule formula to get \(f'(x)\).

Evaluate the derivative at \(x_0 = \frac{1}{2}\) to find the slope of the tangent line: \(m = f'\left(\frac{1}{2}\right)\). Then use the point-slope form of a line: \(y - f\left(\frac{1}{2}\right) = m \left(x - \frac{1}{2}\right)\) to write the equation of the tangent line.

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Derivative and Tangent Line

The derivative of a function at a point gives the slope of the tangent line to the curve at that point. To find the tangent line equation, compute the derivative at x₀ to get the slope, then use the point-slope form y - f(x₀) = f'(x₀)(x - x₀). This line approximates the function near x₀.

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Slopes of Tangent Lines

Quotient Rule for Derivatives

When differentiating a function expressed as a quotient of two functions, use the quotient rule: (u/v)' = (u'v - uv') / v². Here, u = x³ and v = x² + 1, so find u' and v' separately, then apply the rule to find f'(x).

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The Quotient Rule

Evaluating Functions and Derivatives at a Point

After finding the derivative function, substitute the given x₀ value to find the slope at that point. Also, evaluate the original function at x₀ to get the point coordinates. These values are essential to write the tangent line equation accurately.

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Percorso guidato

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Evaluating Composed Functions

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c. Find the slopes of the tangent lines to the graphs of h and k at (2, 2) and (−2, −2).

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In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:

c. Find the equation for the tangent line to f at the specified point (x_0, f(x_0)).

72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2

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