Textbook Question
Owlet talons Let L (t) equal the average length (in mm) of the middle talon on an Indian spotted owlet that is t weeks old, as shown in the figure.<IMAGE>
a. Estimate L' (1.5) and state the physical meaning of this quantity.
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 3.1.26a
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)
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 at a point x = a is given by the limit: \( f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \).
Step 2: Identify the function and the point of interest. Here, the function is \( f(x) = \frac{1}{x} \) and the point P is (1, 1). We need to find \( f'(1) \).
Step 3: Substitute \( f(x) = \frac{1}{x} \) into the derivative formula. This gives: \( f'(1) = \lim_{h \to 0} \frac{f(1+h) - f(1)}{h} = \lim_{h \to 0} \frac{\frac{1}{1+h} - 1}{h} \).
Step 4: Simplify the expression inside the limit. Start by finding a common denominator for the terms in the numerator: \( \frac{1}{1+h} - 1 = \frac{1 - (1+h)}{1+h} = \frac{-h}{1+h} \).
Step 5: Substitute the simplified expression back into the limit: \( f'(1) = \lim_{h \to 0} \frac{-h}{h(1+h)} \). Simplify this to \( \lim_{h \to 0} \frac{-1}{1+h} \) and evaluate the limit as \( h \to 0 \).
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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. For the function f(x) = 1/x, finding the derivative will provide the slope of the tangent line at point P.
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Limit Definition of Derivative
The limit definition of the derivative states that the derivative f'(a) at a point a is the limit of the difference quotient as h approaches zero: f'(a) = lim (h -> 0) [(f(a+h) - f(a))/h]. This definition is fundamental for calculating the slope of the tangent line, as it provides a precise method to determine the instantaneous rate of change at a specific point.
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Related Practice
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.
Textbook Question
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)
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
Textbook Question
Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = √3x; a= 12
Textbook Question
Use the graph of f in the figure to do the following. <IMAGE>
a. Find the values of x in (-2,2) at which f is not continuous.