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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.1.56
Chapter 3, Problem 3.1.56

Find the function The following limits represent the slope of a curve y = f(x) at the point (a,f(a)). Determine a possible function f and number a; then calculate the limit.
(lim x🠂1) 3x²+4x-7 / x-1

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Step 1: Recognize that the given limit represents the derivative of a function f(x) at a point x = a. The expression (lim x→1) (3x² + 4x - 7) / (x - 1) suggests that we are dealing with a derivative at x = 1.
Step 2: Identify the form of the limit. The expression (3x² + 4x - 7) / (x - 1) is in the form of a difference quotient, which is typically used to find the derivative of a function at a specific point.
Step 3: Assume that the function f(x) is a polynomial, since the numerator is a polynomial. A reasonable assumption is that f(x) = 3x² + 4x - 7, which is a quadratic function.
Step 4: To find the derivative f'(x), apply the power rule to each term of the polynomial: f'(x) = d/dx [3x²] + d/dx [4x] - d/dx [7].
Step 5: Evaluate the derivative at x = 1 to find the slope of the tangent line at that point. Substitute x = 1 into f'(x) to calculate the limit.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In this context, the limit helps determine the slope of the curve at a specific point, which is essential for understanding the behavior of the function near that point.
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Derivatives

The derivative of a function at a point gives the slope of the tangent line to the curve at that point. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In this problem, calculating the limit will help us find the derivative of the function at the specified point.
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Derivatives

Function Evaluation

Function evaluation involves substituting a specific value into a function to find its output. In this case, we need to determine a function f and a point a, then evaluate the limit to find the slope at that point. This process is crucial for connecting the limit to the actual behavior of the function.
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Related Practice
Textbook Question

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.

s(t) = 4√t - 1/4t⁴+t+1

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Textbook Question

Given that f(1)=2 and f′(1)=2 , find the slope of the curve y=xf(x) at the point (1, 2).

Textbook Question

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Textbook Question

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.

h(t) = t²/2 + 1

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Textbook Question

Another method for proving lim x→0 cos x−1/x = 0 Use the half-angle formula sin²x = 1− cos 2x/2 to prove that lim x→0 cos x−1/x=0.

Textbook Question

73–78. {Use of Tech} Normal lines A normal line at a point P on a curve passes through P and is perpendicular to the line tangent to the curve at P (see figure). Use the following equations and graphs to determine an equation of the normal line at the given point. Illustrate your work by graphing the curve with the normal line. <IMAGE>


Exercise 48