Problem 31

Question

True or False? Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \text { If } y \text { is differentiable, then } \lim (\Delta y-d y)=0 \text { . } $$

Step-by-Step Solution

Verified

Answer

The statement 'If $y$ is differentiable, then $\lim (\Delta y-d y)=0$ )' is True. This statement is essentially conveying the definition of a derivative at a specific point $x$. The differential $dy$ is an approximation of $\Delta y$ and exactly equal to $\Delta y$ when a function $f(x)$ is both linear and differentiable.

1Step 1: Understand differentiation and the difference and differential

Differentiation, in simple terms, measures the rate at which the value of a function is changing at each point. The difference $\Delta y = f(x+ \Delta x) - f(x)$ and the differential $dy = f'(x) \Delta x$ where $f'(x)$ is the derivative of the function $f(x)$.

2Step 2: Clarify $\Delta y$ versus $dy$

The Difference $\Delta y$ is actually the difference between the value of $y$ at $x + \Delta x$ and the value of $y$ at $x$, whereas the differential $dy$ is the estimate of that difference. The differential $dy$ was created to be an approximation of $\Delta y$. If the function $f(x)$ is linear (which means the graph of the function is a straight line), then $\Delta y = dy$ for each $x$. Otherwise, $\Delta y$ and $dy$ are generally not equal. However, as $\Delta x \to 0$, the approximate equality $\Delta y \approx dy$ becomes the exact equality $\Delta y = dy$.

3Step 3: Connection to Limit

The limit as $\Delta x$ approaches zero of $(\Delta y - dy)$ equals zero is the definition of the derivative, $f'(x)$, of a function at a point $x$. Therefore, the original statement is True. Only when $y$ is differentiable (which means $f(x)$ has a derivative at point $x$), the statement holds true.

Key Concepts

DifferentiationDifferentialsLimits in Calculus

Differentiation

Differentiation lies at the heart of calculus and is a fundamental operation that tells us how a function's output value changes as its input changes. Picture yourself driving a car; differentiation would give you the car's speed at every moment, which is the rate at which your distance from the starting point increases. Mathematically, differentiation is the process of finding the derivative of a function, denoted as $f'(x)$, which measures the instant rate of change of the function at any point $x$.

When we differentiate a function $f(x)$, we're calculating what happens to $y$ as $x$ changes by an infinitesimally small amount. This concept of a tiny change is captured by the derivative, which fundamentally represents the slope of the tangent line to the curve defined by the function at any point on the curve. This slope is how steep the curve is at that point, and thus, the rate at which $y$ is changing with respect to $x$.

Differentials

Differentials, denoted as $dy$, can be thought of as the 'actual change' in response to a small change in $x$, represented by $\text{d}x$. When we say 'actual change', think of it like reading the finest scale on a ruler to measure how much something has grown; it represents a very precise and small change.

The relationship between the differential $dy$ and the change in the function $\text{\Delta y}$ is delicate. The differential $dy$ is an approximation of $\text{\Delta y}$, the true change in $y$, and provides a very close estimate when $\text{\Delta x}$ is small. This is useful in applications like engineering where small changes can be critical, and we need precise calculations.

Understanding $\text{\Delta y}$ and $dy$:

In essence, $\text{\Delta y}$ is the actual difference in $y$ values for two different $x$ values, while $dy$ serves as its linear approximation based on the slope given by the derivative. The smaller the interval for $\text{\Delta x}$, the more accurate the differential $dy$ becomes, virtually mirroring $\text{\Delta y}$ when $\text{\Delta x}$ approaches zero.

Limits in Calculus

Limits in calculus are all about approaching. If you think of walking towards your friend, the limit describes how close you can get without actually reaching out and touching them. Mathematically, a limit examines what happens to a function as the input approaches a certain value, without necessarily ever getting there.

The limit is the mathematical tool that lets us rigorously define derivatives and integrals, the two central concepts in calculus. When we talk about the limit of $(\text{\Delta y} - dy)$ as $\text{\Delta x}$ goes to zero, we're asking, 'What does the difference between the actual change and the estimated change become as the change in $x$ gets tinier and tinier?' If a function is differentiable at a point, that means we can find a limit where this difference actually becomes zero, since the derivative at this point gives us a perfect estimate.

The exercise demonstrates the close relationship between limits and differentiability: a function is differentiable at a point if and only if the limit of the difference between the true change in the function and the estimated change (differential) as the input change approaches zero is itself zero. This provides a powerful insight into the behavior of functions and lays the groundwork for understanding the precision and utility of calculus in modeling the real world.

Problem 31