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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 2.4.68a
Chapter 2, Problem 2.4.68a

Given the graph of f in the following figures, find the slope of the secant line that passes through (0,0) and (h,f(h))in terms of h, for h>0 and h<0.


f(x)=x1/3 <IMAGE>

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1
Step 1: Understand the problem. We need to find the slope of the secant line that passes through the points (0,0) and (h,f(h)) on the graph of the function f(x) = x^{1/3}. The slope of a secant line is given by the formula (f(h) - f(0)) / (h - 0).
Step 2: Calculate f(0). Since f(x) = x^{1/3}, we have f(0) = 0^{1/3} = 0.
Step 3: Calculate f(h). For the function f(x) = x^{1/3}, f(h) = h^{1/3}.
Step 4: Substitute f(0) and f(h) into the slope formula. The slope of the secant line is (h^{1/3} - 0) / (h - 0).
Step 5: Simplify the expression. The slope of the secant line is h^{1/3} / h, which can be further simplified to h^{-2/3} for h > 0 and h < 0.

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Key Concepts

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

Secant Line

A secant line is a straight line that intersects a curve at two or more points. In calculus, it is often used to approximate the slope of the curve between those points. The slope of the secant line can be calculated using the formula (f(b) - f(a)) / (b - a), where a and b are the x-coordinates of the points on the curve.
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Slopes of Tangent Lines

Slope of a Function

The slope of a function at a given point represents the rate of change of the function's value with respect to changes in its input. For a secant line, the slope is determined by the difference in the function's values at two points divided by the difference in their x-coordinates. This concept is foundational for understanding derivatives, which represent instantaneous rates of change.
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Slope-Intercept Form

Cube Root Function

The cube root function, denoted as f(x) = x^(1/3), is a mathematical function that returns the number whose cube is x. This function is defined for all real numbers and has a characteristic shape, being continuous and increasing. Understanding its behavior, especially near the origin, is crucial for analyzing the secant line's slope as h approaches zero from both positive and negative directions.
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Related Practice
Textbook Question

For the following position functions, make a table of average velocities similar to those in Exercises 19–20 and make a conjecture about the instantaneous velocity at the indicated time. 


a. s(t)=−16t^2+80t+60 at t=3

Textbook Question

Determine the following limits.


a. limx→2+1x(x−2){\(\displaystyle\]\lim\)_{x\(\to\)2^{+}}}\(\frac{1}{\sqrt{x\left(x-2\right)}\)}

Textbook Question

Complete the following steps for the given functions. 


a. Find the slant asymptote of ff.


f(x)=x2−2x+53x−2f\(\left\)(x\(\right\))=\(\frac{x^2-2x+5}{3x-2}\)

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Textbook Question

Complete the following steps for the given functions. 


a. Find the slant asymptote of ff.


f(x)=3x2−2x+53x+4f\(\left\)(x\(\right\))=\(\frac{3x^2-2x+5}{3x+4}\)

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Textbook Question

a. Use a graphing utility to estimate lim x→0 tan 2x / sin x, lim x→0 tan 3x / sin x, and lim x→0 tan 4x / sin x.

Textbook Question

A rock is dropped off the edge of a cliff, and its distance s (in feet) from the top of the cliff after t seconds is s(t)=16t^2. Assume the distance from the top of the cliff to the ground is 96 ft.


a. When will the rock strike the ground?