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 7.1.50c

Capitolo 7, Problema 7.1.50c

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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Identify the functions h and k whose tangent line slopes you need to find. Make sure you have their explicit formulas or expressions.

Recall that the slope of the tangent line to a function at a point is given by the derivative of the function evaluated at that point. So, find the derivatives \( h'(x) \) and \( k'(x) \).

Evaluate the derivative \( h'(x) \) at \( x = 2 \) and \( x = -2 \) to find the slopes of the tangent lines to \( h \) at the points \( (2, 2) \) and \( (-2, -2) \).

Similarly, evaluate the derivative \( k'(x) \) at \( x = 2 \) and \( x = -2 \) to find the slopes of the tangent lines to \( k \) at the points \( (2, 2) \) and \( (-2, -2) \).

Summarize the slopes found for each function at the given points, which represent the slopes of the tangent lines to the graphs at those points.

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Derivative as the Slope of the Tangent Line

The derivative of a function at a given point represents the slope of the tangent line to the graph at that point. It measures the instantaneous rate of change of the function with respect to the independent variable.

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Percorso guidato

05:13

Slopes of Tangent Lines

Evaluating the Derivative at Specific Points

To find the slope of the tangent line at a particular point, you first compute the derivative function and then substitute the x-coordinate of the point into this derivative. This yields the slope value at that point.

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Percorso guidato

05:44

Derivatives

Understanding the Graphs of Functions h and k

Knowing the explicit forms or properties of the functions h and k is essential to differentiate them correctly. This understanding allows accurate calculation of their derivatives and evaluation at the given points (2, 2) and (−2, −2).

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Percorso guidato

5:53

Graph of Sine and Cosine Function

Pratica correlata

Domanda del libro di testo

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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:

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