Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.1.16a

Chapter 3, Problem 3.1.16a

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)

Verified step by step guidance

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) = -3x^2 - 5x + 1 $ and the point $ P(1, -7) $. We need to find $ f'(1) $ to determine the slope of the tangent line at $ x = 1 $.

Step 3: Substitute $ a = 1 $ into the limit definition: $ f'(1) = \lim_{h \to 0} \frac{f(1+h) - f(1)}{h} $.

Step 4: Calculate $ f(1) $ by substituting $ x = 1 $ into $ f(x) $: $ f(1) = -3(1)^2 - 5(1) + 1 $.

Step 5: Calculate $ f(1+h) $ by substituting $ x = 1+h $ into $ f(x) $: $ f(1+h) = -3(1+h)^2 - 5(1+h) + 1 $. Expand and simplify this expression.

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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.

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) will provide the slope of the tangent line at point P.

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 at points near P.

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a. For the following functions and values of a, find f′(a).

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Textbook Question

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a. $h prime of 3$

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Textbook Question

21–30. Derivatives

b. Evaluate f'(a) for the given values of a.

f(x) = 1/x+1; a = -1/2;5

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Textbook Question

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b. Determine an equation of the line tangent to the curve at the given point.