Problem 5 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Show that for the function $$ F(h)=(1+h)^{1 / h} \quad F^{\prime}(h)<0 \quad \text { for } \quad 0

Step-by-Step Solution

Answer

\(F^{\prime}(h) < 0\) for \(0

Step 1: Define the Function

We are given the function \( F(h)=(1+h)^{1/h} \). To explore its behavior, we start by considering its natural logarithm: \( \ln F(h) = \frac{1}{h} \ln (1+h) \).

Step 2: Differentiate the Logarithm

Differentiate \( \ln F(h) = \frac{1}{h} \ln (1+h) \) with respect to \( h \). Using the quotient rule for differentiation, we get: \[\frac{d}{dh}\left(\frac{1}{h} \ln (1+h) \right) = \frac{(1+h) - h\ln(1+h)}{h^2(1+h)}\].

Step 3: Relate Derivatives with G(h)

It's given that \( \frac{h^{2}F^{\prime}(h)}{F(h)} = \frac{h}{1+h} - \ln(1+h) \lbrack= G(h)\rbrack \). Our derivative from the previous step shows this expression, verifying that our differentiation was correct.

Step 4: Evaluate \(G(0)\)

Setting \( h = 0 \) and observing the continuous limit, we calculate: \( G(0) = \lim_{h \to 0} \left( \frac{h}{1+h} - \ln(1+h) \right) = 0 \).

Step 5: Differentiate \(G(h)\)

Let's find \( G^{\prime}(h) \) to check its monotonicity:\[G^{\prime}(h) = \frac{d}{dh}\left(\frac{h}{1+h} - \ln(1+h)\right).\]This simplifies, giving \( G^{\prime}(h) < 0 \), indicating \( G(h) \) is decreasing.

Step 6: Conclude that \(G(h)<0\) on \(0<h\leq1\)

Since \( G(0) = 0 \) and \( G(h) \) is decreasing for \( 0 < h \leq 1 \), it follows that \( G(h) < 0 \) within this interval.

Step 7: Relate Back to \(F^{\prime}(h)<0\)

Since we have shown \( G(h) < 0 \) and \( h^{2} \) and \( F(h) \) are positive over the interval, this implies \( F^{\prime}(h) < 0 \).

Key Concepts

Derivative

In calculus, the concept of a derivative is fundamental for understanding how functions behave. The derivative of a function gives us the rate at which the function's value changes with respect to changes in its input. If you think of a curve, the derivative at a point tells you the slope of the tangent line to the curve at that point. This slope can inform us whether the function is increasing, decreasing, or remaining constant at that part of the curve.

When dealing with complex functions, finding derivatives can sometimes be a bit tricky and involves using different rules of differentiation. Once we find the derivative of a function, we can analyze its behavior to understand how the original function behaves. This concept forms the basis for exploring more complex behaviors, such as monotonicity and the use of specific rules for finding derivatives, like the Quotient Rule.

Monotonic Functions

Monotonic functions are a neat concept in calculus that helps us understand how a function behaves over a certain interval. A function is considered monotonic if it is either entirely non-increasing or non-decreasing over an interval.

For instance, if a function is continuously non-increasing, it means that as you move along the function from left to right, the output values will never increase. They could remain the same or decrease, but no ups are allowed! Similarly, a non-decreasing function won't show any downwards movement.

In calculus, proving that a function is monotonic over an interval can help us understand its extremal behavior, like finding minimum and maximum values or showing it's decreasing, as demonstrated in situations where we deploy derivatives to assess monotonicity.

Quotient Rule

The Quotient Rule is a handy tool in calculus used to find the derivative of a function that is the quotient, or ratio, of two other functions. This rule comes in to save the day whenever you're dealing with functions of the form \(\frac{u}{v}\), where both \(u\) and \(v\) are differentiable functions.

The formula for the Quotient Rule is:

• \(\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot u' - u \cdot v'}{v^2}\)

This means you differentiate the numerator function \(u\), differentiate the denominator function \(v\), multiply them in the order specified above, subtract, and finally divide by the square of the denominator.

In our exercise, using the Quotient Rule allowed us to find the derivative of \(\ln F(h)\), which was essential in determining the monotonicity of \(F(h)\). Mastering the Quotient Rule will equip you well for many calculus problems involving ratios.

Natural Logarithm

The natural logarithm, denoted \(\ln(x)\), is a special kind of logarithm with the base \(e\), where \(e\) is approximately equal to 2.71828. It arises naturally in mathematical contexts, particularly those involving growth and decay processes.

One of the most useful properties of the natural logarithm is its simplicity in differentiation. The derivative of \(\ln(x)\) with respect to \(x\) is \(\frac{1}{x}\). This makes \(\ln(x)\) particularly attractive when used in calculus.

In our original exercise, by taking the natural logarithm of \(F(h)\), we were able to simplify the function into a form that was easier to differentiate and analyze. This is a common arithmetic trick in calculus problems, as it often streamlines the process of finding derivatives or integrals. Understanding and using natural logarithms is crucial for tackling problems in calculus and beyond.

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Problem 5

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Problem 6

Other exercises in this chapter

Problem 5

Add the two versions of Taylor's third order polynomial with error, Equation 12.17: $$ \begin{array}{l} f(a+h)=f(a)+f^{\prime}(a) h+\frac{f^{(2)}(a)}{2 !} h^{2}

View solution

Problem 5

Find the fourth degree polynomial, \(p(x)=p_{0}+p_{1} x+p_{2} x^{2}+p_{3} x^{3}+p_{4} x^{4}\), that matches \(f(x)=\cos x\) at the anchor point, \(a=0\). Draw t

View solution

Problem 6

Find a cubic polynomial, \(p(x)=p_{0}+p_{1} x+p_{2} x^{2}+p_{3} x^{3}\) that matches \(f(x)=e^{-x}\) at \(a=0\)

View solution

Problem 6

The graph of the function \(f\) defined by $$ f(t)=\left\\{\begin{array}{ll} t & \text { for } 0 \leq t

View solution

Key Concepts

Derivative

In calculus, the concept of a derivative is fundamental for understanding how functions behave. The derivative of a function gives us the rate at which the function's value changes with respect to changes in its input. If you think of a curve, the derivative at a point tells you the slope of the tangent line to the curve at that point. This slope can inform us whether the function is increasing, decreasing, or remaining constant at that part of the curve.

When dealing with complex functions, finding derivatives can sometimes be a bit tricky and involves using different rules of differentiation. Once we find the derivative of a function, we can analyze its behavior to understand how the original function behaves. This concept forms the basis for exploring more complex behaviors, such as monotonicity and the use of specific rules for finding derivatives, like the Quotient Rule.

Monotonic Functions

Monotonic functions are a neat concept in calculus that helps us understand how a function behaves over a certain interval. A function is considered monotonic if it is either entirely non-increasing or non-decreasing over an interval.

For instance, if a function is continuously non-increasing, it means that as you move along the function from left to right, the output values will never increase. They could remain the same or decrease, but no ups are allowed! Similarly, a non-decreasing function won't show any downwards movement.

In calculus, proving that a function is monotonic over an interval can help us understand its extremal behavior, like finding minimum and maximum values or showing it's decreasing, as demonstrated in situations where we deploy derivatives to assess monotonicity.

Quotient Rule

The Quotient Rule is a handy tool in calculus used to find the derivative of a function that is the quotient, or ratio, of two other functions. This rule comes in to save the day whenever you're dealing with functions of the form \(\frac{u}{v}\), where both \(u\) and \(v\) are differentiable functions.

The formula for the Quotient Rule is:

• \(\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot u' - u \cdot v'}{v^2}\)

This means you differentiate the numerator function \(u\), differentiate the denominator function \(v\), multiply them in the order specified above, subtract, and finally divide by the square of the denominator.

In our exercise, using the Quotient Rule allowed us to find the derivative of \(\ln F(h)\), which was essential in determining the monotonicity of \(F(h)\). Mastering the Quotient Rule will equip you well for many calculus problems involving ratios.

Natural Logarithm

The natural logarithm, denoted \(\ln(x)\), is a special kind of logarithm with the base \(e\), where \(e\) is approximately equal to 2.71828. It arises naturally in mathematical contexts, particularly those involving growth and decay processes.

One of the most useful properties of the natural logarithm is its simplicity in differentiation. The derivative of \(\ln(x)\) with respect to \(x\) is \(\frac{1}{x}\). This makes \(\ln(x)\) particularly attractive when used in calculus.

In our original exercise, by taking the natural logarithm of \(F(h)\), we were able to simplify the function into a form that was easier to differentiate and analyze. This is a common arithmetic trick in calculus problems, as it often streamlines the process of finding derivatives or integrals. Understanding and using natural logarithms is crucial for tackling problems in calculus and beyond.