Problem 57 Define $F(x)$ by $$ F(x)=\... [FREE SOLUTION]

Chapter 5: Problem 57

Define $F(x)$ by $$ F(x)=\int_{1}^{x}\left(3 t^{2}-3\right) d t $$ (a) Use Part 2 of the Fundamental Theorem of Calculus to find $F^{\prime}(x) .$ (b) Check the result in part (a) by first integrating and then differentiating.

Short Answer

Expert verified

(a) $ F'(x) = 3x^2 - 3 $. (b) Confirmed by differentiating $ x^3 - 3x + 2 $.

Step by step solution

Recall the Fundamental Theorem of Calculus (Part 2)

The Fundamental Theorem of Calculus (Part 2) states that if $ F(x) = \int_{a}^{x} f(t) \, dt $, then the derivative $ F'(x) $ is $ f(x) $. For our function, $ f(t) = 3t^2 - 3 $.

Apply the Theorem to Find $ F'(x) $

By the Fundamental Theorem of Calculus, differentiate $ F(x) $: \[ F'(x) = f(x) = 3x^2 - 3 \].

Integrate the Function $ f(t) $

To check the result, first integrate $ f(t) = 3t^2 - 3 $. The antiderivative is obtained by integrating term by term:\[ \int (3t^2 - 3) \, dt = \left( t^3 - 3t \right) + C, \] where $ C $ is the constant of integration.

Apply Limits to the Indefinite Integral

Evaluate the definite integral from 1 to $ x $:\[ \int_{1}^{x} \left( 3t^2 - 3 \right) \, dt = \left[ ( t^3 - 3t) \right]_{1}^{x} = \left( x^3 - 3x \right) - \left( 1^3 - 3 \times 1 \right). \]Simplify:$ x^3 - 3x - (1 - 3) = x^3 - 3x + 2 $.

Differentiate the Resulting Expression

Differentiate $ x^3 - 3x + 2 $ with respect to $ x $:\[ \frac{d}{dx}(x^3 - 3x + 2) = 3x^2 - 3. \]This confirms the derivative $ F'(x) $ found earlier using the Fundamental Theorem.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integration

Integration is a fundamental concept in calculus that deals with finding the antiderivative or the area under a curve. When we integrate a function, we are essentially summing up an infinite number of infinitesimally small contributions to find a total. To integrate a function like $ f(t) = 3t^2 - 3 $, we perform the following steps:

Identify each term in the function and find its individual antiderivative. For example, the antiderivative of $ 3t^2 $ is $ t^3 $ and the antiderivative of $ -3 $ is $ -3t $.
Add these antiderivatives together and include a constant of integration, $ C $, since the process of differentiation would have canceled out constant terms initially.

For $ f(t) $, integrating gives us $ t^3 - 3t + C $. This step reverses differentiation, allowing us to reconstruct the original function from its rate of change.

Derivative

A derivative represents the rate of change of a function. In simpler terms, it tells us how a function's output value changes as its input changes. Calculating the derivative requires us to know the power rule, which involves:

Taking the exponent of $ x $, lowering it by one, and multiplying the initial coefficient by the initial exponent.

For example, in our context with $ F(x) = x^3 - 3x + 2 $, differentiating this with respect to $ x $ would involve using these steps:

The derivative of $ x^3 $ is $ 3x^2 $, as we reduce the power by one and multiply by the original power (3).
The derivative of $ -3x $ is $ -3 $, since we multiply and differentiate with respect to $ x $.
The constant $ 2 $ disappears as the derivative of a constant is zero.

Thus, the derivative of $ F(x) $ leads us back to $ 3x^2 - 3 $, consistent with the fundamental theorem of calculus.

Definite Integral

A definite integral provides the exact value of the total accumulation, such as area under a curve, between two limits, from $ a $ to $ b $. Consider $ \int_{1}^{x} (3t^2 - 3)\, dt $. This involves:

Finding the antiderivative of $ f(t) $, which is $ t^3 - 3t $.
Applying the limits of integration: substituting $ x $ and $ 1 $ into the antiderivative expression.
Subtracting the antiderivative evaluated at $ a $ from that evaluated at $ b $.

The calculation $ [(t^3 - 3t)]_1^x $ results in $ (x^3 - 3x) - (1 - 3) = x^3 - 3x + 2 $.
The definite integral tells us about the net "accumulation" or area between the curve of $ f(t) $ and the $ t $-axis, bounded by $ t = 1 $ and $ t = x $. This is a fundamental tool in many areas of engineering, physics, and mathematical analysis.

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91影视

Define \(F(x)\) by $$ F(x)=\int_{1}^{x}\left(3 t^{2}-3\right) d t $$ (a) Use Part 2 of the Fundamental Theorem of Calculus to find \(F^{\prime}(x) .\) (b) Check the result in part (a) by first integrating and then differentiating.

Short Answer

Step by step solution

Recall the Fundamental Theorem of Calculus (Part 2)

Apply the Theorem to Find \( F'(x) \)

Integrate the Function \( f(t) \)

Apply Limits to the Indefinite Integral

Differentiate the Resulting Expression

Key Concepts

Integration

Derivative

Definite Integral

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