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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 4.2.37
Chapter 4, Problem 4.2.37

Finding Functions from Derivatives


In Exercises 37–40, find the function with the given derivative whose graph passes through the point P.


f'(x) = 2x − 1, P(0,0)

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To find the original function f(x) from its derivative f'(x) = 2x - 1, we need to perform integration. The process of finding a function from its derivative is called antidifferentiation or integration.
Integrate the derivative f'(x) = 2x - 1 with respect to x. This means we need to find the indefinite integral of 2x - 1. The integral of 2x is x^2, and the integral of -1 is -x. Therefore, the integral of f'(x) is: ∫(2x - 1) dx = x^2 - x + C, where C is the constant of integration.
The constant C represents an unknown value that can be determined using the given point P(0,0). Substitute x = 0 and f(x) = 0 into the equation f(x) = x^2 - x + C to find C.
Substituting the point P(0,0) into the equation gives: 0 = 0^2 - 0 + C, which simplifies to C = 0.
Now that we have determined C, the function f(x) is x^2 - x. Therefore, the function whose derivative is 2x - 1 and passes through the point P(0,0) is f(x) = x^2 - x.

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Key Concepts

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

Antiderivatives

An antiderivative of a function is another function whose derivative is the original function. To find a function from its derivative, we perform integration, which is the reverse process of differentiation. In this context, finding the antiderivative of f'(x) = 2x - 1 will yield the original function f(x).
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Antiderivatives

Integration

Integration is the process of finding the antiderivative of a function. It involves determining a function whose derivative is the given function. For f'(x) = 2x - 1, integrating with respect to x will provide the general form of f(x), which includes an arbitrary constant C.
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Introduction to Indefinite Integrals

Initial Conditions

Initial conditions are used to find the specific solution to a differential equation by determining the constant of integration. Given the point P(0,0), we substitute x = 0 and f(x) = 0 into the integrated function to solve for C, ensuring the function passes through the specified point.
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Initial Value Problems
Related Practice
Textbook Question

Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.

8. y = 2cosx - √2x, -π≤x≤3π/2

Textbook Question

26. Constructing cylinders Compare the answers to the following two construction problems.

a. A rectangular sheet of perimeter 36 cm and dimensions x cm by y cm is to be rolled into a cylinder as shown in part (a) of the figure. What values of x and y give the largest volume?

b. The same sheet is to be revolved about one of the sides of length y to sweep out the cylinder as shown in part (b) of the figure. What values of x and y give the largest volume?

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Textbook Question

107. Marginal cost The accompanying graph shows the hypothetical cost c=f(x) of manufacturing x items. At approximately what production level does the marginal cost change from decreasing to increasing?

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Textbook Question

Theory and Examples


In Exercises 53 and 54, show that the function has neither an absolute minimum nor an absolute maximum on its natural domain.


y = x¹¹ + x³ + x − 5

Textbook Question

10. Catching rainwater A 1125 ft^3 open-top rectangular tank with a square base x ft on a side and y ft deep is to be built with its top flush with the ground to catch runoff water. The costs associated with the tank involve not only the material from which the tank is made but also an excavation charge proportional to the product xy.

a. If the total cost is c=5(x^2+4xy) + 10xy, what values of x and y will minimize it?

b. Give a possible scenario for the cost function in part (a).

Textbook Question

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.


∫(−3csc²x)dx