Problem 31 - Calculus Early Transcendentals: Pearson New International Edition

Question

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 $$ F(t)=\int_{\sin \sqrt{2} \pi t}^{t^{2}} \sqrt{9+u^{4}} d u $$

Step-by-Step Solution

Answer

\(F'(\sqrt{2}) = 10\sqrt{2}\).

Step 1: Understand the Problem

We are given a function defined by an integral with variable limits, \(F(t) = \int_{g(t)}^{h(t)} f(u) \, du\), where \(g\) and \(h\) are differentiable, and \(f\) is continuous. We need to find the derivative \(F'(t)\).

Step 2: Use Leibniz Rule

According to Leibniz Rule for differentiation under the integral sign, for a function of the form \(F(t) = \int_{g(t)}^{h(t)} f(u) \, du\), the derivative is given by \[ F'(t) = f(h(t)) h'(t) - f(g(t)) g'(t) \]

Step 3: Identify Functions and Derivatives

For the function \(F(t) = \int_{\sin(\sqrt{2} \pi t)}^{t^2} \sqrt{9+u^4} \, du\): - \(g(t) = \sin(\sqrt{2} \pi t)\) with derivative \(g'(t) = \sqrt{2} \pi \cos(\sqrt{2} \pi t)\).- \(h(t) = t^2\) with derivative \(h'(t) = 2t\).- \(f(u) = \sqrt{9+u^4}\), a function of \(u\) only, no derivative needed.

Step 4: Apply Leibniz Rule for F'(t)

Substitute \(g(t)\), \(h(t)\) and their derivatives into Leibniz rule: \[ F'(t) = \sqrt{9+(t^2)^4}\cdot 2t - \sqrt{9+\sin^4(\sqrt{2} \pi t)} \cdot \sqrt{2} \pi \cos(\sqrt{2} \pi t) \] which simplifies to \[ F'(t) = \sqrt{9+t^8}\cdot 2t - \sqrt{9+\sin^4(\sqrt{2} \pi t)} \cdot \sqrt{2} \pi \cos(\sqrt{2} \pi t) \]

Step 5: Evaluate F' at t = \(\sqrt{2}\)

Substitute \(t = \sqrt{2}\) into the expression for \(F'(t)\):- \(h(t) = (\sqrt{2})^2 = 2\) so \(\sqrt{9+2^4} \cdot 2\sqrt{2} = \sqrt{9+16} \cdot 2\sqrt{2} = 5 \cdot 2\sqrt{2}\).- \(g(t) = \sin(\sqrt{2} \pi \sqrt{2}) = \sin(2\pi) = 0\) so the second term becomes \(0\).Therefore, \[ F'(\sqrt{2}) = 10\sqrt{2} \]

Step 6: Conclusion

The derivative \(F'(t)\) evaluates to \(10\sqrt{2}\) when \(t = \sqrt{2}\).

Key Concepts

Variable Limits Integration

When working with integrals where the limits themselves are functions of a variable, we refer to this scenario as integration with variable limits. In these cases, the limits of integration are not constant values, but functions such as \( g(t) \) and \( h(t) \), making the entire integral dependent on a variable \( t \).

This introduces a dynamic element to the integration, as changes in the variable \( t \) alter the range over which the integration is performed. Variable limits integration is particularly useful in scenarios where you wish your integration bounds to reflect some changing condition or state, dependent on another varying parameter.

• Leads to a dependency of the integral on \( t \), often requiring differentiation with respect to \( t \).
• Common in problems involving areas and probabilities that parameterize over time or other variables.

Understanding variable limits integration is crucial for further application in problems that require differentiation under the integral sign.

Differentiation Under the Integral Sign

Differentiation under the integral sign, often associated with Leibniz Rule, is a powerful technique for evaluating derivatives of integral-defined functions.

This technique allows you to differentiate an integral where the limits of integration or the integrand itself depends on a variable. The general form, when applying the Leibniz Rule, is:
\[ F'(t) = f(h(t)) h'(t) - f(g(t)) g'(t) \]
Here, \( f \) is the integrand, while \( g(t) \) and \( h(t) \) are the variable limits, both differentiable with respect to \( t \). The derivative is found by computing the value of \( f \) at the upper limit multiplied by the derivative of the upper limit, minus \( f \) computed at the lower limit multiplied by the derivative of the lower limit.

• Effectively uses the continuity of \( f \) and the differentiability of limits.
• Makes complex integral-based functions more approachable in calculus.

This method is crucial when analyzing physical processes or mathematical models involving rates of change influenced by progressing variables.

Continuous Function

A continuous function is one that does not have any abrupt changes in value – it is smooth and unbroken. Continuity of a function, in the context of integration in calculus, ensures that the integral behaves predictably as conditions change.

For a function \( f(u) \) to be continuous over a given interval, it must satisfy:

• The function is defined at every point in the interval.
• There are no jumps, gaps, or vertical asymptotes within the interval.
• Limits as you approach any point in the interval from either direction equal the function's value at that point.

Continuity is crucial because:

• It ensures the Fundamental Theorem of Calculus is applicable, allowing integration and differentiation to interact seamlessly.
• Establishes a foundation for predicting the outcome of applying Leibniz Rule, as the function's smooth nature assures that variations in the integral can be differentiated.

When working with integrals involving continuous functions, like in the problem of variable limits integration, we rely on this property to accurately calculate derivatives and solve complex real-life scenarios.

Differentiable Functions

For functions \( g(t) \) and \( h(t) \) used as variable boundaries in integration, being differentiable means they have a well-defined derivative at all points within their interval of interest.

A differentiable function is essentially one that can be smoothly traced without sharp corners or cusps. In calculus, differentiability is a stricter condition than continuity, although every differentiable function is continuous.

• Differentiability provides insights into the function's rate of change and smoothness.
• It ensures the possibility of determining slopes or direction at any given point on the function.

Specifically, when values like \( g(t) \) or \( h(t) \) serve as the limits of integration, their differentiability assures that:

• We can calculate \( g'(t) \) and \( h'(t) \) for applying the Leibniz Rule.
• It guarantees the error-free evaluation of rate-related concepts in applied math, physics, and engineering.

Understanding differentiation and differentiability allows us to handle variable limits adeptly, leading to more complex and insightful analyses of system behaviors or geometric transformations.