Textbook Question
Derivative Calculations
In Exercises 1–12, find the first and second derivatives.
y = x² + x + 8
Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 3, Problem 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.
Verified step by step guidance1
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.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
Recommended video:
05:45
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.
Recommended video:
05:13Slopes 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.
Recommended video:
05:45Understanding Slope Fields
Related Practice
Textbook Question
Slopes, Tangent Lines, and Normal Lines
In Exercises 31–40, verify that the given point is on the curve and find the lines that are (a) tangent and (b) normal to the curve at the given point.
y = 2 sin(πx – y), (1,0)
Textbook Question
If L = √(x² + y²), dx/dt = –1, and dy/dt = 3, find dL/dt when x = 5 and y = 12.
Textbook Question
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.)
<IMAGE>
Textbook Question
In Exercises 29 and 30, find the slope of the curve at the given points.
(x² + y²)² = (x – y)² at (1,0) and (1,–1)
Textbook Question
Falling meteorite The velocity of a heavy meteorite entering Earth’s atmosphere is inversely proportional to √s when it is s km from Earth’s center. Show that the meteorite’s acceleration is inversely proportional to s².