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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 2.1.29a
Chapter 2, Problem 2.1.29a

Tangent lines with zero slope


a. Graph the function f(x)=x^2−4x+3.

Verified step by step guidance
1
We need to graph the function \( f(x) = x^2 - 4x + 3 \) and identify where the tangent lines have a zero slope.
The derivative of \( f(x) \), denoted as \( f'(x) \), gives us the slope of the tangent line at any point \( x \). Calculate \( f'(x) \) using the power rule.
To find where the tangent line has a zero slope, set \( f'(x) = 0 \) and solve for \( x \).
Solve the equation from Step 3 to find the x-coordinate(s) where the slope of the tangent line is zero.
Plot the function \( f(x) = x^2 - 4x + 3 \) and mark the point(s) found in Step 4 on the graph. These are the points where the tangent line is horizontal.>

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Key Concepts

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

Tangent Lines

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. Understanding tangent lines is crucial for analyzing the behavior of functions, especially in relation to their derivatives.
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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) = x^2 - 4x + 3, finding the derivative will help identify points where the slope of the tangent line is zero, indicating potential local maxima or minima.
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Derivatives

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax^2 + bx + c. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the coefficient 'a'. Analyzing the graph of f(x) = x^2 - 4x + 3 will reveal its vertex, intercepts, and the nature of its tangent lines.
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Introduction to Polynomial Functions
Related Practice
Textbook Question

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a- f(x) and lim x→a+ f(x).

f(x) = (x^4−1)/(x^2−1)

Textbook Question

Complete the following steps for the given functions. 


a. Find the slant asymptote of ff.


f(x)=4x3+4x2+7x+4x2+1f\(\left\)(x\(\right\))=\(\frac{4x^3+4x^2+7x+4}{x^2+1}\)

Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.

The line x=−1 is a vertical asymptote of the function f(x) =x^2 − 7x + 6 / x^2 − 1.

1
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Textbook Question

The hyperbolic cosine function, denoted cosh(x)\(\cosh\]\left\)(x\(\right\)), is used to model the shape of a hanging cable (a telephone wire, for example). It is defined as cosh(x)=ex+e−x2\(\cosh\]\left\)(x\(\right\))=\(\frac{e^{x}\)+e^{-x}}{2}.


b. Evaluate cosh(0)\(\cosh\]\left\)(0\(\right\)). Use symmetry and part (a) to sketch a plausible graph for y=cosh(x)y=\(\cosh\]\left\)(x\(\right\)).

Textbook Question

A projectile is fired vertically upward and has a position given by s(t)=−16t^2+128t+192, for 0≤t≤9.


a. Graph the position function, for 0≤t≤9.

Textbook Question

Let g(x)=x3−4x8∣x−2∣g\(\left\)(x\(\right\))=\(\frac{x^3-4x}{8\left|x-2\right|}\). <IMAGE>

Calculate g(x)g\(\left\)(x\(\right\)) for each value of xx in the following table.