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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 3 - DerivativesProblema 3.1.3
Capitolo 3, Problema 3.1.3

Slopes and Tangent Lines


In Exercises 1–4, use the grid and a straight edge to make a rough estimate of the slope of the curve (in y-units per x-unit) at the points P₁ and P₂.
Graph showing a curve with points P₁ and P₂ marked, illustrating slopes and tangent lines on a grid.

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1
Identify the points P₁ and P₂ on the graph. P₁ is located at approximately (0.5, 1) and P₂ is at approximately (2, 1.5).
To estimate the slope at P₁, draw a tangent line at this point. A tangent line is a straight line that touches the curve at exactly one point and represents the instantaneous rate of change at that point.
Determine the slope of the tangent line at P₁. The slope is calculated as the change in y divided by the change in x (rise over run). Use the grid to estimate these changes.
Repeat the process for point P₂. Draw a tangent line at P₂ and estimate its slope using the grid.
Compare the slopes at P₁ and P₂ to understand how the rate of change of the curve varies between these two points.

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Slope of a Curve

The slope of a curve at a given point represents the rate of change of the function at that point. It is calculated as the ratio of the change in the y-coordinate to the change in the x-coordinate (rise over run) as you approach the point. For curves, this is often approximated using secant lines between two points, which can give insight into the behavior of the curve.
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Understanding Slope Fields

Tangent Line

A tangent line to a curve at a specific point is a straight line that just touches the curve at that point, representing the instantaneous rate of change of the function. The slope of the tangent line is equal to the derivative of the function at that point. This concept is crucial for understanding how functions behave locally around specific points.
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Slopes of Tangent Lines

Estimating Slopes

Estimating slopes involves using graphical methods to approximate the slope of a curve at specific points. This can be done by drawing a tangent line at the point of interest and measuring its steepness. In the context of the given question, students are asked to visually assess the slope at points P₁ and P₂ using a grid, which helps develop intuition about derivatives and rates of change.
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Understanding Slope Fields
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Derivatives in Differential Form


In Exercises 17–28, find dy.


y = (2√x)/(3(1 + √x))

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If L = √(x² + y²), dx/dt = –1, and dy/dt = 3, find dL/dt when x = 5 and y = 12.

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Cylinder pressure If gas in a cylinder is maintained at a constant temperature T, the pressure P is related to the volume V by a formula of the form

P = (nRT / (V − nb)) − (an² / V²),

in which a, b, n, and R are constants. Find dP/dV. (See accompanying figure.)


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Find the derivatives of the functions in Exercises 19–40.


g(t) = (1 + sin(3t) / (3 − 2t))⁻¹

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Find the derivatives of the functions in Exercises 19–40.


y = (5 − 2x)⁻³ + (1 / 8)(2 / x + 1)⁴

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One-Sided Derivatives


Compute the right-hand and left-hand derivatives as limits to show that the functions in Exercises 37–40 are not differentiable at the point P.

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