Ch. 4 - Applications of Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problema 4.1.27

Capitolo 4, Problema 4.1.27

Absolute Extrema on Finite Closed Intervals

In Exercises 21–36, find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.

h(x) = ³√x, −1 ≤ x ≤ 8

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First, understand that to find the absolute extrema (maximum and minimum values) of a function on a closed interval, we need to evaluate the function at critical points and endpoints of the interval.

Identify the critical points of the function h(x) = ³√x. To do this, find the derivative of h(x) and set it equal to zero. The derivative of h(x) = ³√x is h'(x) = (1/3)x^(-2/3).

Since h'(x) = (1/3)x^(-2/3) is never zero, there are no critical points where the derivative is zero. However, check where the derivative is undefined. In this case, h'(x) is undefined at x = 0, which is within the interval [-1, 8].

Evaluate the function h(x) at the endpoints of the interval and at any critical points found. Calculate h(-1), h(0), and h(8).

Compare the values obtained from evaluating h(x) at x = -1, x = 0, and x = 8 to determine the absolute maximum and minimum values. These values will indicate the absolute extrema on the interval [-1, 8].

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Absolute Extrema

Absolute extrema refer to the highest and lowest values a function attains on a given interval. To find these, evaluate the function at critical points and endpoints of the interval. The absolute maximum is the largest value, and the absolute minimum is the smallest value among these evaluations.

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Finding Extrema Graphically

Critical Points

Critical points occur where the derivative of a function is zero or undefined. These points are potential locations for local extrema. For the function h(x) = ³√x, the derivative is undefined at x = 0, making it a critical point to consider when finding absolute extrema on the interval.

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Critical Points

Graphing Functions

Graphing a function helps visualize its behavior over an interval, including where it reaches its extrema. For h(x) = ³√x, plot the function from x = -1 to x = 8, marking the critical points and endpoints. This visual representation aids in identifying the coordinates of the absolute maximum and minimum values.

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Graph of Sine and Cosine Function

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Finding Position from Velocity or Acceleration

Exercises 45–48 give the acceleration a=d²s/dt², initial velocity, and initial position of an object moving on a coordinate line. Find the object’s position at time t.

a = 9.8, v(0) = −3, s(0) = 0

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A cone is formed from a circular piece of material of radius 1 meter by removing a section of angle θ and then joining the two straight edges. Determine the largest possible volume for the cone.

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Finding Extrema from Graphs

In Exercises 7–10, find the absolute extreme values and where they occur.

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Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫(1 + tan²θ)dθ (Hint:1 + tan²θ = sec²θ)

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Roots (Zeros)

Show that the functions in Exercises 19–26 have exactly one zero in the given interval.

f(x) = x⁴ + 3x + 1, [−2, −1]

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Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫csc θ/(csc θ − sin θ) dθ

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