Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
1. c. tan^(-1)(1/√3)
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
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).
Verified step by step guidance1
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.
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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 exact location.
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05:44Derivatives
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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6:02Determining Different Coordinates for the Same Point
Related Practice
Textbook Question
10. True, or false? As x→∞,
c. 1/x - 1/x² = o(1/x)
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
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)).
67. y= √(3x-2), 2/3 ≤ x ≤ 4, x_0=3
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)).
68. y= (3x+2)/(2x-11), -2 ≤ x ≤ 2, x_0=1/2
Textbook Question
23. Human evolution continues The analysis of tooth shrinkage by C. Loring Brace and colleagues at the University of Michigan’s Museum of Anthropology indicates that human tooth size is continuing to decrease and that the evolutionary process has not yet come to a halt. In northern Europeans, for example, tooth size reduction now has a rate of 1% per 1000 years.
c. What will be our descendants’ tooth size 20,000 years from now (as a percentage of our present tooth size)?