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Chapter 4: Problem 24

Compute \(F(c)\) from the given information. $$ F^{\prime}(x)=1 / x, F(1)=3, c=-e^{2} $$

Short Answer

Expert verified
\( F(-e^2) = 5 \).

Step by step solution

01

Recognize the Derivative Function

We are given that the derivative of the function is \( F'(x) = \frac{1}{x} \). This means that the function \( F(x) \) is the antiderivative of \( \frac{1}{x} \), which is \( \ln|x| + C \) where \( C \) is a constant of integration.
02

Use Initial Condition to Find C

We know \( F(1) = 3 \). Substituting in \( F(x) = \ln|x| + C \), we have \( F(1) = \ln|1| + C = 3 \). Since \( \ln(1) = 0 \), it follows that \( C = 3 \). Thus, \( F(x) = \ln|x| + 3 \).
03

Evaluate F at c

Now that we have \( F(x) = \ln|x| + 3 \), we need to evaluate \( F(c) \) where \( c = -e^2 \). Since \( |c| = e^2 \), substitute to get \( F(-e^2) = \ln(e^2) + 3 \).
04

Simplify the Expression

Calculate \( \ln(e^2) = 2 \ln(e) = 2 \times 1 = 2 \). Thus, \( F(-e^2) = 2 + 3 = 5 \).

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Key Concepts

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

Derivative Functions
When you hear the term "derivative," think of it as a tool that tells you how a function changes at any given point. In this exercise, we're dealing with the derivative function ul>
  • Derivative Function: Given by \( F'(x) = \frac{1}{x} \), this tells us the rate of change of the original function \( F(x) \) with respect to \( x \).
  • Antiderivative: To find the original function (called the antiderivative), we determine what could have been differentiated to become \( F'(x) \). Here, the antiderivative of \( \frac{1}{x} \) is the natural logarithm function, \( \ln|x| \).
    • The beauty of derivative functions lies in their ability to reverse-engineer or backtrack, giving us the original function when combined with initial conditions.
    Constant of Integration
    In calculus, every time we find an antiderivative, we add a constant of integration. This constant is crucial, because derivatives do not reveal it:
    • Why Add It?: The derivative of a constant is zero, so any real number added to \( \ln|x| \) would differentiate back to \( \frac{1}{x} \) and thus we add \( C \) to account for any such "invisible" constants.
    • Expression: Once we have the antiderivative, we write it as \( F(x) = \ln|x| + C \).
    By finding \( C \), we customize the antiderivative to fit particular conditions or data points provided in a problem.
    Initial Conditions
    To find any unknown constants in the antiderivative, we apply initial conditions from the problem:
    • Use the Given Data Point: We were told \( F(1) = 3 \). Using \( F(x) = \ln|x| + C \), we substitute to find the specific value of \( C \).
    • Solving for C: Plug \( x = 1 \) into the equation to get \( 3 = \ln(1) + C \). Since \( \ln(1) = 0 \), we determine that \( C = 3 \).
    Initial conditions help anchor our function to real situations, allowing us to solve concrete problems accurately.

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    Most popular questions from this chapter

    Darboux's Theorem (after Jean Darboux, \(1842-1917\) ) states that if \(f^{\prime}\) exists at every point of an open interval containing \([a, b],\) and if \(\gamma\) is between \(f^{\prime}(a)\) and \(f^{\prime}(b),\) then there is a \(c\) in the interval \((a, b)\) such that \(f^{\prime}(c)=\gamma .\) The existence of such a \(c\) follows from the Intermediate Value Theorem if \(f^{\prime}\) is assumed continuous. Darboux's Theorem, which you will prove in this exercise, tells us that the assumption of continuity is not needed for functions that are derivatives. Suppose for definiteness that \(f^{\prime}(a)<\gamma<\) \(f^{\prime}(b) .\) Define \(g\) on \([a, b]\) by \(g(x)=f(x)-\gamma x\) a. Explain why \(g\) has a minimum value that occurs at some point \(c\) in \((a, b)\). b. Show that \(f^{\prime}(c)=\gamma\). c. Use Darboux's Theorem to show that if \(f\) increases on the interval \((l, c)\) and decreases on the interval \((c, r)\) (or vice versa), then \(f^{\prime}(c)=0\)

    Approximate the critical points and inflection points of the given function \(f\). Determine the behavior of \(f\) at each critical point. $$ f(x)=\tanh (x)-x^{2} /\left(x^{2}+1\right) $$

    In each of Exercises 54-60, determine each point \(c,\) where the given function \(f\) satisfies \(f^{\prime}(c)=0\). At each such point, use the First Derivative Test to determine whether \(f\) has a local maximum, a local minimum, or neither. $$ f(x)=\tanh (x)-\exp (2 x) $$

    In each of Exercises 82-89, use the first derivative to determine the intervals on which the function is increasing and on which the function is decreasing. $$ f(x)=x^{4}-x^{2}-7.1 x+3.2 $$

    Use the Chain Rule to calculate the given indefinite integral. $$ \int x\left(x^{2}+1\right)^{100} d x $$
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