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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1:27 minutes

Problem 3.1.43

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

Verified step by step guidance

Step 1: Begin by understanding the concept of a vertical tangent line. A vertical tangent occurs where the derivative of the function is undefined or infinite. For the function y = 4x^(2/5) - 2x, we need to find its derivative.

Step 2: Differentiate the function y = 4x^(2/5) - 2x with respect to x. Use the power rule for differentiation: if y = x^n, then dy/dx = n*x^(n-1). Apply this to each term in the function.

Step 3: The derivative of y = 4x^(2/5) is (2/5)*4*x^((2/5)-1) = (8/5)*x^(-3/5). The derivative of -2x is simply -2. Therefore, the derivative of the function is dy/dx = (8/5)*x^(-3/5) - 2.

Step 4: Identify where the derivative dy/dx = (8/5)*x^(-3/5) - 2 is undefined or infinite. This occurs when x^(-3/5) is undefined, which happens when x = 0. Thus, x = 0 is a candidate for a vertical tangent line.

Step 5: Confirm the vertical tangent by calculating the limit of the derivative as x approaches 0. Evaluate the limit of (8/5)*x^(-3/5) - 2 as x approaches 0 from the left and right. If the limit approaches infinity or negative infinity, a vertical tangent exists at x = 0.

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Key Concepts

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

Vertical Tangent Lines

Vertical tangent lines occur at points on a curve where the derivative is undefined or infinite. This typically happens when the slope of the tangent line becomes vertical, indicating a sharp turn or cusp in the graph. Identifying these points requires analyzing the behavior of the derivative as it approaches certain values.

Limit Calculations

Limit calculations are used to determine the behavior of a function as it approaches a specific point. In the context of vertical tangents, limits help confirm where the derivative becomes infinite or undefined. Calculating limits involves evaluating the function's behavior as the input approaches the point of interest, often using algebraic manipulation or L'Hôpital's Rule.

Derivative of a Function

The derivative of a function represents the rate of change or slope of the function at any given point. For the function y = 4x²/⁵ − 2x, finding the derivative involves applying differentiation rules, such as the power rule, to each term. The derivative is crucial for identifying tangent lines and understanding the function's behavior, especially in determining where vertical tangents may occur.

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Textbook Question

Interpreting Derivative Values

Effectiveness of a drug On a scale from 0 to 1, the effectiveness E of a pain-killing drug t hours after entering the bloodstream is displayed in the accompanying figure.

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a. At what times does the effectiveness appear to be increasing? What is true about the derivative at those times?

Textbook Question

Find an equation of the straight line having slope 1/4 that is tangent to the curve y = √x.

Textbook Question

Rates of Change

Speed of a rocket At t sec after liftoff, the height of a rocket is 3t² ft. How fast is the rocket climbing 10 sec after liftoff?

Textbook Question

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

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

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.

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[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

Key Concepts

Vertical Tangent Lines

Limit Calculations

Derivative of a Function

Watch next

[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