Ch. 7 - Transcendental Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problem 7.1.50c

Chapter 7, Problem 7.1.50c

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

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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Related Practice

Textbook Question

Suppose that the function f and its derivative with respect to x have the following values at x=0, 1, 2, 3, and 4.

Assuming the inverse function f^(-1) is differentiable, find the slope of f^(-1)(x) at

c. x=3

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Textbook Question

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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Textbook Question

9. True, or false? As x→∞,

c. x = O(x+5)

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Textbook Question

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

5. c. arccos(√3/2)

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Textbook Question

In Exercises 1–4, solve for t.

1. c. e^((ln 0.2)t) = 0.4

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Textbook Question

4. Use the properties of logarithms to write the expressions in Exercises 3 and 4 as a single term.

c. 3ln ∛(t² - 1) - ln(t+1)

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