Problem 10
Question
If \(z=f(x)+y g(x),\) what can we say about \(z_{y y} ?\) \(\odot z_{y y}=0\) \(\odot z_{y y}=y\) \(\odot z_{y y}=z_{x x}\) \(\odot z_{y y}=g(x)\) \(\odot\) We cannot say anything
Step-by-Step Solution
Verified Answer
\[z_{yy}=0\]
1Step 1: Find \(z_y\)
To find the first derivative of \(z\) with respect to \(y\), take the partial derivative of the given function with respect to y:
\[z_y=\frac{\partial}{\partial y} (f(x)+y g(x))\]
Since \(f(x)\) does not depend on \(y\), its derivative with respect to \(y\) is 0. Thus, we get:
\[z_y=0+\frac{\partial}{\partial y} (y g(x)) = g(x)\]
#Step 2: Find the second derivative of z with respect to y#
2Step 2: Find \(z_{yy}\)
To find the second derivative of \(z\) with respect to \(y\), take the partial derivative of \(z_y\) with respect to y:
\[z_{yy}=\frac{\partial}{\partial y} (g(x))\]
Since \(g(x)\) does not depend on \(y\), its derivative with respect to \(y\) is 0. Therefore, we have:
\[z_{yy}=0\]
Now, let us compare our result with the options given.
#Step 3: Compare the result to the options#
3Step 3: Compare the result
The result we found for \(z_{yy}\) is 0, which matches the option \(\odot z_{yy}=0\). Therefore, we can say that \(z_{yy}=0\) is the correct statement about the second derivative of \(z\) with respect to \(y\).
Key Concepts
Partial DerivativesMultivariable CalculusSecond DerivativeDifferentiation
Partial Derivatives
Understanding partial derivatives is critical when studying functions with more than one variable. In essence, a partial derivative measures how a function changes as one of its variables is altered, while the other variables are held constant. This is like observing how the surface of a hill changes in slope as you walk in a straight line north, ignoring changes in the east-west direction.
For example, given a function like \( z = f(x) + y g(x) \), we identify that \( z \) is dependent on two variables, \( x \) and \( y \). To find the partial derivative of \( z \) with respect to \( y \), denoted as \( z_y \), we treat \( x \) as a constant and differentiate only the terms that involve \( y \). In this instance, the term \( f(x) \) is independent of \( y \), thus its partial derivative with respect to \( y \) is zero.
In the given exercise, we accurately calculated \( z_y = g(x) \) by applying this principle. The partial derivative can tell us not just about the rate of change, but also about the behavior of the function with respect to one variable, illuminating the function's multidimensional nature.
For example, given a function like \( z = f(x) + y g(x) \), we identify that \( z \) is dependent on two variables, \( x \) and \( y \). To find the partial derivative of \( z \) with respect to \( y \), denoted as \( z_y \), we treat \( x \) as a constant and differentiate only the terms that involve \( y \). In this instance, the term \( f(x) \) is independent of \( y \), thus its partial derivative with respect to \( y \) is zero.
In the given exercise, we accurately calculated \( z_y = g(x) \) by applying this principle. The partial derivative can tell us not just about the rate of change, but also about the behavior of the function with respect to one variable, illuminating the function's multidimensional nature.
Multivariable Calculus
Multivariable calculus expands on the principles of calculus to functions of several variables, rather than just one. It's like upgrading from studying the contours of a line to exploring the vast landscape of a mountain range. This branch of mathematics is essential for analyzing systems where multiple factors are at play simultaneously.
In the case of our example function, \( z = f(x) + y g(x) \), we're dealing with a function that depends on two variables, which makes it a perfect candidate for multivariable calculus techniques. Multivariable calculus employs concepts like partial derivatives, gradient vectors, and multiple integrals to study these functions. It allows us to calculate rates of change in several dimensions and understand how these changes in one variable can affect others. An important aspect of multi-variable calculus is recognizing that each partial derivative only gives information about one aspect of the function's variation.
In the case of our example function, \( z = f(x) + y g(x) \), we're dealing with a function that depends on two variables, which makes it a perfect candidate for multivariable calculus techniques. Multivariable calculus employs concepts like partial derivatives, gradient vectors, and multiple integrals to study these functions. It allows us to calculate rates of change in several dimensions and understand how these changes in one variable can affect others. An important aspect of multi-variable calculus is recognizing that each partial derivative only gives information about one aspect of the function's variation.
Second Derivative
The second derivative provides us with information about the curvature of a function's graph—essentially, it tells us whether the function is 'bending' upwards or downwards, and how sharply it's doing so. In the context of multivariable calculus, we encounter partial second derivatives, such as \( z_{yy} \), which indicates the rate at which the first derivative \( z_y \) with respect to \( y \) is changing as \( y \) changes.
In the exercise, we found that the second derivative of \( z \) with respect to \( y \), which we write as \( z_{yy} \), is zero. This implies that the slope given by the first derivative \( z_y \) is constant with respect to \( y \), suggesting there is no curvature along the \( y \)-axis. When \( z_{yy} = 0 \), it's like walking along a straight path on our metaphorical hill—no matter how far you walk in the \( y \)-direction, your elevation doesn't change.
In the exercise, we found that the second derivative of \( z \) with respect to \( y \), which we write as \( z_{yy} \), is zero. This implies that the slope given by the first derivative \( z_y \) is constant with respect to \( y \), suggesting there is no curvature along the \( y \)-axis. When \( z_{yy} = 0 \), it's like walking along a straight path on our metaphorical hill—no matter how far you walk in the \( y \)-direction, your elevation doesn't change.
Differentiation
Differentiation is the process of finding the derivative, which measures how a quantity changes in response to changes in another quantity. This foundational tool in calculus can be thought of as finding the instantaneous rate of change or the slope of the tangent line at any point on a curve.
When differentiating functions of a single variable, we get a single derivative. However, with functions of multiple variables, such as in our exercise, we make use of partial differentiation to investigate the effect of changing one variable at a time. Differentiation, whether partial or not, reveals much about the behavior of functions: where they increase or decrease, where they have maxima or minima, and how they interact with other variables. As showcased in our step-by-step solution of the exercise, differentiation (and in particular, partial differentiation) acted as a surgical tool to dissect the function and examine how it alters with respect to \( y \) alone, ultimately determining that the effect of \( y \) doesn't bring about variation in the slope of \( z \).
When differentiating functions of a single variable, we get a single derivative. However, with functions of multiple variables, such as in our exercise, we make use of partial differentiation to investigate the effect of changing one variable at a time. Differentiation, whether partial or not, reveals much about the behavior of functions: where they increase or decrease, where they have maxima or minima, and how they interact with other variables. As showcased in our step-by-step solution of the exercise, differentiation (and in particular, partial differentiation) acted as a surgical tool to dissect the function and examine how it alters with respect to \( y \) alone, ultimately determining that the effect of \( y \) doesn't bring about variation in the slope of \( z \).
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