Problem 52 - Calculus for Biology and Medicine

Question

(a) Graph \(y=x^{5 / 2}, x \geq 0\), and \(y=x^{1 / 2}, x \geq 0\), together, in one coordinate system. (b) Show algebraically that $$ x^{5 / 2} \leq x^{1 / 2} $$ for \(0 \leq x \leq 1\). (Hint: Show that \(x^{1 / 2} / x^{-1 / 2}=x \leq 1\) for \(0

Step-by-Step Solution

Answer

(a) Graph shows where one function is above another. (b) For \(0 < x \leq 1\), \(x^{5/2} \leq x^{1/2}\). (c) For \(x \geq 1\), \(x^{5/2} \geq x^{1/2}\).

Step 1: Graph Aquistion

To graph the functions \( y = x^{5/2} \) and \( y = x^{1/2} \) on the same coordinate system for \( x \geq 0 \), we need to observe the basic shapes of these functions. The function \( y = x^{1/2} \) is a square root function, which is concave downward and passes through the points like \( (0,0), (1,1), (4,2) \). The function \( y = x^{5/2} \) grows faster than \( y = x^{1/2} \) for \( x > 1 \) because it increases at a faster rate due to the higher exponent, but initially, for \( 0 < x < 1 \), it is less than or equal to \( y = x^{1/2} \). Plot the line \( y=x \) to observe the intersection points.

Step 2: Analyzing Inequality for 0≤x≤1

Starting with the hint, we'll show \( \frac{x^{1/2}}{x^{-1/2}} = x \). Simplifying the fraction gives \( x^{1/2} \times x^{1/2} = x \), which equals \( x \). Since \( 0 \leq x \leq 1 \) clearly indicates that \( x \leq 1 \), this implies that for \( 0 < x \leq 1 \):\[x^{5/2} = (x^{1/2})^5 = (x^{1/2})(x^{1/2})^2(x^{1/2}) \]is less than or equal to \( x^{1/2} \) because \( x^{1/2} \times x \leq x^{1/2} \times 1 \). This means \( x^{5/2} \leq x^{1/2} \) when \( 0

Step 3: Analyzing Inequality for x≥1

For \( x \geq 1 \), we note that raising \( x \) to a higher power increases its value, i.e., \( x^{5/2} \geq x^{1/2} \). Writing \( x^{5/2} \) as \( (x^{1/2})^5 \), we can see:\[x^{5/2} = x^{1/2} imes x^2 \]As \( x \geq 1 \), then \( x^2 \geq x \geq 1 \), therefore \( x^{5/2} \geq x \geq x^{1/2} \). Hence, when \( x \geq 1 \), \( x^{5/2} \geq x^{1/2} \).

Key Concepts

Inequalities

An inequality is a mathematical statement that shows the relationship of one quantity being greater or less than another. For expressions like those in our exercise,

• "\( x^{5/2} \leq x^{1/2} \) for \( 0 \leq x \leq 1 \)" means that for all values of \( x \) within the range from 0 to 1, the value of \( x^{5/2} \) is either less than or equal to \( x^{1/2} \).
• Conversely, "\( x^{5/2} \geq x^{1/2} \) for \( x \geq 1 \)" indicates that starting from \( x = 1 \), \( x^{5/2} \) will always be greater than or equal to \( x^{1/2} \).

Understanding inequalities allows us to predict how equations behave over certain intervals.**Using Inequalities in Practice**- We solve these inequalities by manipulating and transforming the inequalities algebraically, ensuring we retain their validity.- By analyzing how functions compare under different conditions, we can identify intervals where one overtakes the other.- This is an excellent way to predict the behavior of a function without graphing.

Exponents

Exponents are a shorthand for repeated multiplication. If you see \( x^{n} \), it means that \( x \) is multiplied by itself \( n \) times. In our exercise:- The exponent \( \frac{5}{2} \) implies taking the squareroot followed by raising the result to the power of 5.- The expression \( x^{1/2} \) represents the square root of \( x \).**Properties of Exponents**

• Combining exponents, such as when multiplying like bases, involves adding their exponents: \( x^{a} \times x^{b} = x^{a+b} \).
• Power of a power involves multiplying: \( (x^{a})^{b} = x^{a\cdot b} \).

Exponents enable us to solve complex equations by breaking them into more manageable parts. They also help us understand the rate of growth or decline of functions over different sections of their domain.

Functions

A function is essentially a rule that assigns each input exactly one output. In this exercise, we are dealing with functions \( y = x^{5/2} \) and \( y = x^{1/2} \). Understanding these helps to:- Predict the output of a range of inputs.- Analyze their algebraic properties like monotonicity or concavity.**Key Properties**- **Domain and Range**: For \( y = x^{5/2} \) and \( y = x^{1/2} \), both are defined where \( x \geq 0 \). The range for \( x \geq 0 \) for both functions is non-negative real numbers.- **Concavity**: \( y = x^{1/2} \) is concave down while \( y = x^{5/2} \) turns upward more steeply for larger \( x \), showing different growth behaviors.

• Concave down indicates that the slope is decreasing as \( x \) increases.
• For \( y = x^{5/2} \), larger exponents make the function grow faster over larger \( x \).

Functions like these are commonly used to model and solve real-world problems involving growth rates, optimization, and understanding natural patterns.

Graphing

Graphing is a visual method to understand functions and their relationships. By graphing the functions \( y = x^{5/2} \) and \( y = x^{1/2} \) on the same coordinate system, we can observe their interactions and compare their growth visually.**Graphing Benefits**

• Provides a clear, intuitive understanding of how functions increase or decrease over their domain.
• Identifies key points of intersection where the functions are equal - particularly useful in exercises that involve inequalities.

When plotting these specific functions:- Start by calculating a few key points for each function such as \( x=0, 1, 4 \), and plot them.- Notice how \( y = x^{1/2} \) which is the square root function, grows more slowly compared to \( y = x^{5/2} \) as \( x \) increases. It starts to curve downward, illustrating its concavity.- By looking closely, you can see \( x^{5/2} \)'s faster growth rate as it curves upwards as \( x \geq 1 \).Graphing provides an essential insight into understanding relationships within calculus, particularly with regard to inequalities, showcasing the effect of different exponents. It strengthens the comprehension of mathematical predictions visually.

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Key Concepts

Inequalities

An inequality is a mathematical statement that shows the relationship of one quantity being greater or less than another. For expressions like those in our exercise,

• "\( x^{5/2} \leq x^{1/2} \) for \( 0 \leq x \leq 1 \)" means that for all values of \( x \) within the range from 0 to 1, the value of \( x^{5/2} \) is either less than or equal to \( x^{1/2} \).
• Conversely, "\( x^{5/2} \geq x^{1/2} \) for \( x \geq 1 \)" indicates that starting from \( x = 1 \), \( x^{5/2} \) will always be greater than or equal to \( x^{1/2} \).

Understanding inequalities allows us to predict how equations behave over certain intervals.**Using Inequalities in Practice**- We solve these inequalities by manipulating and transforming the inequalities algebraically, ensuring we retain their validity.- By analyzing how functions compare under different conditions, we can identify intervals where one overtakes the other.- This is an excellent way to predict the behavior of a function without graphing.

Exponents

Exponents are a shorthand for repeated multiplication. If you see \( x^{n} \), it means that \( x \) is multiplied by itself \( n \) times. In our exercise:- The exponent \( \frac{5}{2} \) implies taking the squareroot followed by raising the result to the power of 5.- The expression \( x^{1/2} \) represents the square root of \( x \).**Properties of Exponents**

• Combining exponents, such as when multiplying like bases, involves adding their exponents: \( x^{a} \times x^{b} = x^{a+b} \).
• Power of a power involves multiplying: \( (x^{a})^{b} = x^{a\cdot b} \).

Exponents enable us to solve complex equations by breaking them into more manageable parts. They also help us understand the rate of growth or decline of functions over different sections of their domain.

Functions

A function is essentially a rule that assigns each input exactly one output. In this exercise, we are dealing with functions \( y = x^{5/2} \) and \( y = x^{1/2} \). Understanding these helps to:- Predict the output of a range of inputs.- Analyze their algebraic properties like monotonicity or concavity.**Key Properties**- **Domain and Range**: For \( y = x^{5/2} \) and \( y = x^{1/2} \), both are defined where \( x \geq 0 \). The range for \( x \geq 0 \) for both functions is non-negative real numbers.- **Concavity**: \( y = x^{1/2} \) is concave down while \( y = x^{5/2} \) turns upward more steeply for larger \( x \), showing different growth behaviors.

• Concave down indicates that the slope is decreasing as \( x \) increases.
• For \( y = x^{5/2} \), larger exponents make the function grow faster over larger \( x \).

Functions like these are commonly used to model and solve real-world problems involving growth rates, optimization, and understanding natural patterns.

Graphing

Graphing is a visual method to understand functions and their relationships. By graphing the functions \( y = x^{5/2} \) and \( y = x^{1/2} \) on the same coordinate system, we can observe their interactions and compare their growth visually.**Graphing Benefits**

• Provides a clear, intuitive understanding of how functions increase or decrease over their domain.
• Identifies key points of intersection where the functions are equal - particularly useful in exercises that involve inequalities.

When plotting these specific functions:- Start by calculating a few key points for each function such as \( x=0, 1, 4 \), and plot them.- Notice how \( y = x^{1/2} \) which is the square root function, grows more slowly compared to \( y = x^{5/2} \) as \( x \) increases. It starts to curve downward, illustrating its concavity.- By looking closely, you can see \( x^{5/2} \)'s faster growth rate as it curves upwards as \( x \geq 1 \).Graphing provides an essential insight into understanding relationships within calculus, particularly with regard to inequalities, showcasing the effect of different exponents. It strengthens the comprehension of mathematical predictions visually.