Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.1.24a

Chapter 3, Problem 3.1.24a

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.
f(x) = 8 - 2x²; P(0, 8)

Verified step by step guidance

Identify the function f(x) = 8 - 2x^2 and the point P(0, 8) where you need to find the slope of the tangent line.

Recall the definition of the derivative as the slope of the tangent line at a point: f'(x) = lim_{h \(\to\) 0} \(\frac{f(x+h) - f(x)}{h}\).

Substitute f(x) = 8 - 2x^2 into the derivative definition: f'(x) = lim_{h \(\to\) 0} \(\frac{(8 - 2(x+h)^2) - (8 - 2x^2)}{h}\).

Simplify the expression inside the limit: f'(x) = lim_{h \(\to\) 0} \(\frac{-2(x^2 + 2xh + h^2) + 2x^2}{h}\).

Further simplify and evaluate the limit as h approaches 0 to find f'(x), then substitute x = 0 to find the slope of the tangent line at P(0, 8).

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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 around the point of interest.

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Related Practice

Textbook Question

7–14. Find the derivative the following ways:

a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.

g(s) = 4s³ - 8s² +4s / 4s

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Textbook Question

If two opposite sides of a rectangle increase in length, how must the other two opposite sides change if the area of the rectangle is to remain constant?

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Textbook Question

Consider the following cost functions.

a. Find the average cost and marginal cost functions.

C(x) = 1000+0.1x, 0≤x≤5000, a=2000

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Textbook Question

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

(x+y)^2/3=y; (4, 4)