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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.3.68a
Chapter 4, Problem 4.3.68a

Theory and Examples


Sketch the graph of a differentiable function y = f(x) that has a local minimum at (1, 1) and a local maximum at (3, 3).

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Start by understanding the behavior of the function around the local minimum and maximum. A local minimum at (1, 1) means that the derivative changes from negative to positive at x = 1, indicating a 'valley' in the graph. Similarly, a local maximum at (3, 3) means the derivative changes from positive to negative at x = 3, indicating a 'peak'.
Consider the general shape of the graph. Between x = 1 and x = 3, the function should increase to reach the local maximum at (3, 3). Before x = 1 and after x = 3, the function should decrease, reflecting the local minimum and maximum respectively.
Sketch the graph starting from the left of x = 1. The function should decrease until it reaches the point (1, 1). At this point, the slope of the tangent line is zero, indicating a local minimum.
From x = 1 to x = 3, sketch the graph such that it increases to reach the local maximum at (3, 3). At this point, the slope of the tangent line is zero again, indicating a local maximum.
Finally, sketch the graph beyond x = 3, where the function should decrease again. Ensure the graph reflects the change in slope at the local minimum and maximum points, creating a smooth curve that is differentiable everywhere.

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Key Concepts

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

Differentiable Function

A differentiable function is one that has a derivative at every point in its domain. This means the function is smooth and continuous, without any sharp corners or cusps. Understanding differentiability is crucial for sketching graphs, as it ensures the function's behavior can be predicted using its derivative.
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Finding Differentials

Local Minimum and Maximum

A local minimum is a point where the function value is lower than all nearby points, while a local maximum is where the function value is higher than all nearby points. These points are critical in graph sketching, as they indicate where the function changes direction, often corresponding to where the derivative equals zero.
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The Second Derivative Test: Finding Local Extrema

Critical Points and Derivatives

Critical points occur where the derivative of a function is zero or undefined, indicating potential local minima or maxima. Analyzing the derivative helps determine the nature of these points, guiding the sketching of the function's graph by showing where it increases or decreases.
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Critical Points
Related Practice
Textbook Question

Checking Antiderivative Formulas


Right, or wrong? Say which for each formula and give a brief reason for each answer.


∫√(2x + 1) dx = √(x² + x +C)

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Textbook Question

Theory and Examples


In Exercises 51 and 52, give reasons for your answers.


Let f(x) = |x³ − 9x|.


a. Does f'(0) exist?

Textbook Question

52. Two masses hanging side by side from springs have positions s_1 = 2 sin t and s_2 = sin 2t,

respectively.

a. At what times in the interval 0 < t do the masses pass each other? (Hint: sin 2t = 2 sint cost.)

Textbook Question

Finding displacement from an antiderivative of velocity


a. Suppose that the velocity of a body moving along the s-axis is


ds/dt = v = 9.8t − 3.


i. Find the body’s displacement over the time interval from t = 1 to t = 3 given that s = 5 when t = 0.

Textbook Question

Checking Antiderivative Formulas


Right, or wrong? Say which for each formula and give a brief reason for each answer.


∫tanθ sec²θ dθ = sec³θ / 3 + C

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Textbook Question

Identifying Extrema


In Exercises 41–52:


a. Identify the function’s local extreme values in the given domain, and say where they occur.


f(x) = √(x² − 2x − 3), 3 ≤ x < ∞