Ch. 3 - Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 3 - Derivatives

Problem 3.1.24a

Chapter 3, Problem 3.1.24a

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.

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

Verified step by step guidance

Step 1: Observe the graph of the effectiveness E versus time t. The effectiveness E increases from t = 0 to t = 3, as the curve slopes upward during this interval.

Step 2: Recall that the derivative of a function represents the rate of change of the function. When the effectiveness E is increasing, the derivative dE/dt is positive.

Step 3: Identify the interval where the slope of the graph is positive. From the graph, the slope is positive from t = 0 to t = 3, indicating that the derivative dE/dt > 0 during this interval.

Step 4: Note that at t = 3, the graph reaches its maximum point, where the slope becomes zero. This means the derivative dE/dt = 0 at t = 3.

Step 5: Conclude that the effectiveness appears to be increasing during the interval 0 < t < 3, and the derivative is positive during this time.

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Key Concepts

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

Derivative and Increasing Function

The derivative of a function at a point gives the slope of the tangent line at that point. If the derivative is positive over an interval, the function is increasing on that interval. In the context of the drug's effectiveness, the derivative E'(t) is positive when the effectiveness is increasing, which can be observed from the graph where the slope is upward.

Critical Points and Maximum Effectiveness

Critical points occur where the derivative is zero or undefined, often indicating potential maxima or minima. In the graph, the effectiveness reaches a maximum when the derivative changes from positive to negative, which is around t = 3 hours. This is where the drug's effectiveness is at its peak before it starts to decrease.

Interpreting Graphs in Calculus

Understanding how to interpret graphs is crucial in calculus. The graph of E(t) shows the effectiveness of the drug over time. By analyzing the slope of the graph, one can determine intervals of increase or decrease. The graph indicates that the effectiveness increases from t = 0 to t = 3 and decreases thereafter, which aligns with the derivative's behavior.

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Related Practice

Textbook Question

Find y'' if:

a. y = csc x

Textbook Question

Interpreting Derivative Values

Growth of yeast cells In a controlled laboratory experiment, yeast cells are grown in an automated cell culture system that counts the number P of cells present at hourly intervals. The number after t hours is shown in the accompanying figure.

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a. Explain what is meant by the derivative P'(5). What are its units?

Textbook Question

Differentiability and Continuity on an Interval

Each figure in Exercises 45–50 shows the graph of a function over a closed interval D. At what domain points does the function appear to be

a. differentiable?

Give reasons for your answers.

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Textbook Question

If the original 24 m edge length x of a cube decreases at the rate of 5 m/min, when x = 3 m at what rate does the cube’s

a. surface area change?

Textbook Question

Differentiability and Continuity on an Interval

Each figure in Exercises 45–50 shows the graph of a function over a closed interval D. At what domain points does the function appear to be

a. differentiable?

Give reasons for your answers.

Textbook Question

Differentiability and Continuity on an Interval

Each figure in Exercises 45–50 shows the graph of a function over a closed interval D. At what domain points does the function appear to be

a. differentiable?

Give reasons for your answers.