Ch. 7 - Transcendental Functions

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

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problem 7.1.49c

Chapter 7, Problem 7.1.49c

c. Find the slopes of the tangent lines to the graphs of f and g at (1, 1) and (−1, −1) (four tangent lines in all).

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Identify the functions \( f(x) \) and \( g(x) \) whose tangent line slopes you need to find. The problem refers to points \((1, 1)\) and \((-1, -1)\), so ensure you have the explicit forms of \( f(x) \) and \( g(x) \) before proceeding.

Find the derivatives \( f'(x) \) and \( g'(x) \) of the functions \( f(x) \) and \( g(x) \) respectively. The derivative represents the slope of the tangent line at any point \( x \). Use differentiation rules such as the power rule, product rule, or chain rule as needed.

Evaluate the derivative \( f'(x) \) at \( x = 1 \) to find the slope of the tangent line to \( f \) at the point \( (1, 1) \). Similarly, evaluate \( f'(x) \) at \( x = -1 \) to find the slope at \( (-1, -1) \).

Evaluate the derivative \( g'(x) \) at \( x = 1 \) to find the slope of the tangent line to \( g \) at the point \( (1, 1) \). Then evaluate \( g'(x) \) at \( x = -1 \) to find the slope at \( (-1, -1) \).

Summarize the four slopes obtained: two from \( f'(x) \) at \( x = 1 \) and \( x = -1 \), and two from \( g'(x) \) at the same points. These represent the slopes of the tangent lines to the graphs of \( f \) and \( g \) at the given 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 exact location.

Understanding the Coordinates of Points on the Graph

The points given, such as (1, 1) and (−1, −1), lie on the graphs of the functions f and g. Confirming these points satisfy the function equations ensures the tangent lines are correctly evaluated at valid points on the curves.

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Related Practice

Textbook Question

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

1. c. tan^(-1)(1/√3)

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Textbook Question

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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Textbook Question

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

7. c. arcsec(-2)

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Textbook Question

In Exercises 41–44:

c. Evaluate df/dx at x = a and df⁻¹/dx at x = f(a) to show that

(df⁻¹/dx)|ₓ₌f(a) = 1 / (df/dx)|ₓ₌a

44. f(x) = 2x², x ≥ 0, a = 5

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Textbook Question

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

68. y= (3x+2)/(2x-11), -2 ≤ x ≤ 2, x_0=1/2

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Textbook Question

5. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?

c. ln(√x)

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