Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

2. Intro to Derivatives

Tangent Lines and Derivatives

Calculus

2. Intro to Derivatives

Tangent Lines and Derivatives

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2:16 minutes

Problem 3.1.1

Textbook Question

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 guidance

To estimate the slope of the curve at point P₁, first identify the coordinates of P₁ on the graph. It appears that P₁ is at (0, 0).

Next, draw a tangent line at point P₁ using a straight edge. The tangent line should just touch the curve at P₁ without crossing it.

Estimate the slope of the tangent line at P₁ by choosing two points on the tangent line. Calculate the rise over run (change in y over change in x) between these two points.

Repeat the process for point P₂. Identify the coordinates of P₂, which appear to be approximately (1, 2) on the graph.

Draw a tangent line at point P₂ using a straight edge. Estimate the slope of this tangent line by selecting two points on it and calculating the rise over run.

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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 vertical change (rise) to the horizontal change (run) between two points on the curve. For a smooth curve, this slope can be approximated by drawing a tangent line at the point of interest.

Tangent Line

A tangent line to a curve at a specific point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line at that point is equal to the derivative of the function at that point, providing a linear approximation of the curve's behavior nearby.

Estimating Slopes

Estimating slopes involves visually analyzing the graph to determine the steepness of the curve at specific points. This can be done by selecting two points close to the point of interest and calculating the slope of the line connecting them, or by drawing a tangent line and measuring its slope. This estimation is crucial for understanding the function's behavior in calculus.

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Related Practice

Textbook Question

Show that the line y = mx + b is its own tangent line at any point (x₀, mx₀ + b).

Textbook Question

[Technology Exercise]

Graph the curves in Exercises 39–48.

a. Where do the graphs appear to have vertical tangent lines?

b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38.

y = 4x²/⁵ − 2x

Textbook Question

Tangent line to y = √x Does any tangent line to the curve y = √x cross the x-axis at x = −1? If so, find an equation for the line and the point of tangency. If not, why not?

Textbook Question

In Exercises 19–22, find the slope of the curve at the point indicated.

y = (x − 1) / (x + 1), x = 0

Textbook Question

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.

Textbook Question

In Exercises 5–10, find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.

y = 4 − x², (−1, 3)

Textbook Question

In Exercises 5–10, find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.

y = (1 / x²), (−1, 1)

Textbook Question

In Exercises 5–10, find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.

y = (1 / x³), (−2, −1/8)

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Slopes and Tangent LinesIn 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.

Key Concepts

Slope of a Curve

Tangent Line

Estimating Slopes

Watch next

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.