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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 7 - Transcendental FunctionsProblema 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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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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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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5:53
Graph of Sine and Cosine Function
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Domanda del libro di testo

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

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3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?

c. √(x^4 + x^3)

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Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.

2. b. tan^(-1)(√3)

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1. Express the following logarithms in terms of ln 2 and ln 3.

b. ln(4/9)

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75. b. Identify the function’s local and absolute extreme values, if any, saying where they occur.

g(x) = x(ln x)²

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Domanda del libro di testo

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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