Textbook Question
{Use of Tech} A damped oscillator The displacement of a mass on a spring suspended from the ceiling is given by .
a. Graph the displacement function.
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 3.1.22a
Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.
f(x) = -7x; P(-1,7)
Verified step by step guidance1
Step 1: Recall the definition of the derivative as the slope of the tangent line at a point. The derivative of a function \( f(x) \) at a point \( x = a \) is given by \( f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \).
Step 2: Identify the function \( f(x) = -7x \) and the point \( P(-1, 7) \). Here, \( a = -1 \) and \( f(a) = 7 \).
Step 3: Substitute \( f(x) = -7x \) into the derivative definition: \( f'(a) = \lim_{h \to 0} \frac{f(-1+h) - f(-1)}{h} \).
Step 4: Calculate \( f(-1+h) \) and \( f(-1) \). Since \( f(x) = -7x \), we have \( f(-1+h) = -7(-1+h) = 7 - 7h \) and \( f(-1) = -7(-1) = 7 \).
Step 5: Substitute these values into the limit expression: \( f'(a) = \lim_{h \to 0} \frac{(7 - 7h) - 7}{h} \). Simplify the expression and evaluate the limit to find the slope of the tangent line.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Tangent Line
A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line represents the instantaneous rate of change of the function at that point, which is crucial for understanding how the function behaves locally.
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Slopes of Tangent Lines
Derivative
The derivative of a function at a point quantifies how the function's output changes as its input changes. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In this context, finding the derivative of f(x) = -7x will provide the slope of the tangent line at point P.
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Point-Slope Form
The point-slope form of a linear equation is used to express the equation of a line given a point on the line and its slope. It is written as y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. This form is useful for constructing the equation of the tangent line once the slope is determined.
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3:56Slope-Intercept Form
Related Practice
Textbook Question
62–65. {Use of Tech} Graphing f and f'
a. Graph f with a graphing utility.
f(x) = (x−1) sin^−1 x on [−1,1]
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Textbook Question
31–32. Velocity functions A projectile is fired vertically upward into the air, and its position (in feet) above the ground after t seconds is given by the function s(t).
a. For the following functions s(t), find the instantaneous velocity function v(t). (Recall that the velocity function v is the derivative of the position function s.)
s(t)= −16t²+100t
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Textbook Question
Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.
f(x) = 1/x; P (1,1)
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Textbook Question
Vertical tangent lines If a function f is continuous at a and lim x→a| f′(x)|=∞, then the curve y=f(x) has a vertical tangent line at a, and the equation of the tangent line is x=a. If a is an endpoint of a domain, then the appropriate one-sided derivative (Exercises 71–72) is used. Use this information to answer the following questions.
73. {Use of Tech} Graph the following functions and determine the location of the vertical tangent lines.
a. f(x) = (x-2)^1/3
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Textbook Question
Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.
f(x) = 2/√x; P(4,1)
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