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Magazine Chapter 12: Problem 31
Let \(F(t)=\int_{g(t)}^{h(t)} f(u) d u,\) where \(f\) is continuous and \(g\) and \(h\) are differentiable. Show that $$ F^{\prime}(t)=f(h(t)) h^{\prime}(t)-f(g(t)) g^{\prime}(t) $$ and use this result to find \(F^{\prime}(\sqrt{2}),\) where $$ F(t)=\int_{\sin \sqrt{2} \pi t}^{t^{2}} \sqrt{9+u^{4}} d u $$
Short Answer
Expert verified
\( F'(\sqrt{2}) = \sqrt{41} \cdot 2\sqrt{2} \).
Step by step solution
01
Understand the Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus, Part 1, states that if \( F(t) = \int_{a}^{b} f(u) du \), where \( a \) and \( b \) are constants, then \( F'(t) = 0 \) because \( F \) is constant. However, if the limits are functions of \( t \), as is the case here, we have to use the Fundamental Theorem of Calculus for differentiation under the integral sign. This requires additional steps involving the Chain Rule.
02
Differentiate the Integral with Respect to \( t \)
To differentiate \( F(t)=\int_{g(t)}^{h(t)} f(u) du \), we use the Leibniz Rule, which states that \( F'(t) = f(h(t)) \cdot h'(t) - f(g(t)) \cdot g'(t) \). This formula accounts for the changing limits of integration.
03
Identify the Components in \( F(t) = \int_{\sin \sqrt{2} \pi t}^{t^2} \sqrt{9+u^4} du \)
Here, \( g(t) = \sin \sqrt{2} \pi t \) and \( h(t) = t^2 \). Thus, \( f(u) = \sqrt{9 + u^4} \). We need to find \( f(h(t)) \), \( f(g(t)) \), \( h'(t) \), and \( g'(t) \).
04
Calculate Derivatives \( h'(t) \) and \( g'(t) \)
Compute \( h'(t) = \frac{d}{dt} (t^2) = 2t \) and \( g'(t) = \frac{d}{dt} (\sin \sqrt{2} \pi t) = \sqrt{2} \pi \cos(\sqrt{2} \pi t) \).
05
Apply the Chain Rule and Leibniz Rule to Find \( F'(t) \)
Using the Leibniz Rule, plug in the derivatives and the function values: \[ F'(t) = \sqrt{9 + (t^2)^4} \cdot 2t - \sqrt{9 + (\sin \sqrt{2} \pi t)^4} \cdot \sqrt{2} \pi \cos(\sqrt{2} \pi t) \]
06
Evaluate \( F'( \sqrt{2}) \)
Substitute \( t = \sqrt{2} \) into the expression obtained in Step 5. Calculate \[ F'(\sqrt{2}) = \sqrt{9 + (\sqrt{2})^8} \cdot 2\sqrt{2} - \sqrt{9 + (\sin \sqrt{2} \pi \cdot \sqrt{2})^4} \cdot \sqrt{2} \pi \cos(\sqrt{2} \pi \cdot \sqrt{2}) \].Simplify the expression to find the final value for \( F'(\sqrt{2}) \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus is a bridge between differentiation and integration, two main operations in calculus. This theorem has two parts. The first part states that if you have a continuous function and you integrate it over an interval, differentiating this integral function will return the original function. However, in our scenario, since the integral limits are not constants but functions of another variable, we apply the second part of the theorem which incorporates differentiation under the integral sign. This allows us to differentiate the integral when the limits are not constant, but are themselves functions. Therefore, understanding this theorem is crucial for working through such integrals, especially when employing differentiation and other calculus techniques.
Differentiation
Differentiation is the process of finding the derivative of a function. The derivative represents how a function's output value changes as its input value changes. In our problem, we need to differentiate an integral function when the limits of integration are functions rather than constants. This calls for differentiation under the integral sign, a technique informed by the Fundamental Theorem of Calculus. Here, instead of finding the derivative of a basic function, you must determine how the integral changes in respect to the variable and its varying limits. This often involves additional calculus rules such as the Chain Rule and Leibniz Rule, making it a step-by-step process for function evaluation.
Chain Rule
The Chain Rule is a method in calculus for differentiating composite functions. When you have a function inside another function, like in our integral limits, the Chain Rule helps you differentiate them effectively. In simple terms, it allows you to "chain" the derivative of the outside function with the derivative of the inside function. For example, if one of our integration limits is a function of another variable, applying the Chain Rule helps us find how the change in this limit affects the original function. It ensures that the changes to multiple levels are correctly accounted for, an essential step for solving the problem like the one given.
Leibniz Rule
The Leibniz Rule is a handy tool for differentiating an integral where the limits themselves are functions of a variable, say, t. It is especially useful for our problem where the integral's upper and lower bounds are not constants but functions. The formula summarizing Leibniz Rule is based on both the Chain Rule and the Fundamental Theorem of Calculus. It specifies that the derivative of such an integral considers how the integrand changes and how the limits' values change. Specifically:
- Apply part of the rule to differentiate the composite functions at the limits.
- Take the derivative of the function at the upper limit, multiplied by the derivative of the upper limit.
- Subtract the derivative of the function at the lower limit, multiplied by the derivative of the lower limit.
