Textbook Question
Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.
f(x) = -3x2 - 5x + 1; P(1,-7)
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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 3.1.25a
Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.
f(x) = x2 - 4; P(2, 0)
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 \( f(x) = x^2 - 4 \) and the point P is (2, 0). We need to find \( f'(2) \).
Step 3: Substitute \( a = 2 \) into the derivative definition: \( f'(2) = \lim_{h \to 0} \frac{f(2+h) - f(2)}{h} \).
Step 4: Calculate \( f(2+h) \) and \( f(2) \). We have \( f(2+h) = (2+h)^2 - 4 \) and \( f(2) = 2^2 - 4 \).
Step 5: Simplify the expression \( \frac{f(2+h) - f(2)}{h} \) and evaluate the limit as \( h \to 0 \) 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) = x² - 4 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 using the function's values around the point of interest.
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