Question

Writing to Learn Recall that the area \(A\) of a circle with radius \(r\) is \(\pi r^{2}\) and that the circumference \(C\) is 2\(\pi r\) . Notice that \(d A / d r=C .\) Explain in terms of geometry why the instantaneous rate of change of the area with respect to the radius should equal the circumference.

Short answer

The rate of change of the area of a circle with respect to the radius equals its circumference because an infinitesimal increase in the radius adds an additional 'ring' or 'strip' of area that is precisely the length of the circumference of the circle.

Step-by-step solution

Definition of Area and Circumference

For a circle, the area \(A\) is given by \(\pi r^{2}\), and the circumference \(C\) is given by \(2\pi r\). These are the basic definitions that we know from geometry.

Understand the Concept of Derivative

A derivative measures how a function changes as its input changes. In this case, it's about how the area \(A\) changes as the radius \(r\) changes. The notation \(dA / dr\) is the derivative of the area with respect to the radius.

Derive the Area

The derivative of \(A\) with respect to \(r\) is \(dA / dr = 2\pi r\). This is calculated using the power rule of derivatives where the derivative of \(x^n\) is \(nx^{n-1}\). Here, the power on \(r\) is 2, so the derivative is \(2\pi r\).

Comparing Derivative with the Circumference

We notice that the derivative of the area with respect to the radius equals the circumference \(C = 2\pi r\).

Geometrical Interpretation

This means that if we increase the radius by a very small amount, the increase in the area will be equivalent to wrapping the additional circumference created around the existing circle. In other words, the instantaneous rate of change of the area of the circle is equal to the circumference because every time you increase the radius (even infinitesimally), the increase in area is exactly the length of the additional circumference added.

Key concepts

Instantaneous Rate of Change

Understanding the instantaneous rate of change is akin to capturing a snapshot of an object's speed at a specific moment in time. Imagine steadily increasing the radius of a circle; as you do so, the area of that circle changes continuously. The instantaneous rate of change, in this context, is how swiftly the area increases as the radius expands precisely at that moment. This concept is a fundamental aspect of calculus, sometimes referred to as the derivative.

In the case of a circle's area, when looking at a very small change in radius, we can determine how much the area changes in response. This is visually represented by the new sliver of the area that appears between the original circle and the new, slightly larger one. The relationship between this instantaneous change in area and the circle's circumference is deeply entwined, which leads to a compelling geometrical connection revealed through calculus.

By connecting the dots between instantaneous rate of change and real-life scenarios, like the spread of ripples in a pond when a stone is dropped in, we can appreciate the power of calculus to describe dynamic systems.

Geometry and Calculus Integration

When we merge the visual world of geometry with the dynamic world of calculus, fascinating insights emerge. Geometry tells us about shapes, sizes, and the properties of space. Calculus, on the other hand, deals with change and motion. Integrating these two fields allows for a deeper comprehension of both.

For the circle, the integration of geometry and calculus manifests when we relate the area and circumference with calculus concepts like derivatives. Suddenly, these static geometrical measures become dynamic, providing us with more information about how they behave in relation to one another as they change. This harmonious integration not only enriches our understanding of mathematics but also equips us with the tools to solve complex problems in physics, engineering, and beyond.

Linking geometry's fixed values, such as the circle's area at a given radius, with calculus's focus on change, illuminates the beautiful mechanics underlying the shapes we often take for granted.

Circumference of a Circle

The circumference of a circle is the distance around its edge, which is unequivocally linked with its radius. It's expressed with the familiar formula: \( C = 2\pi r \). This simple yet elegant relationship forms a cornerstone in both geometry and calculus.

In terms of the physical world, if you were to take a string and lay it out along the circumference of a circle, then measure the length of this string, you'd have the circle's circumference. As the radius changes, so too does the circumference in direct proportion. This directly observable behavior exemplifies the concept of the circumference and highlights the interconnectedness of different aspects of a circle's geometry, proving essential for understanding more advanced mathematics such as calculus.

Power Rule of Derivatives

The power rule of derivatives is a quick, efficient means to find the rate at which a quantity changes. In this rule, if you have a function of the form \( x^n \) where \( n \) is any real number, the derivative of that function with respect to \( x \) is \( nx^{n-1} \). It's a fundamental tool of differentiation in calculus.

Applied to our circle scenario, the area of the circle is a function of its radius \( r \) raised to the power of 2 (\( \pi r^2 \) — where the 2 represents the square). Using the power rule, we differentiate \( r^2 \) to get \( 2r \) and then multiply by \( \pi \) to account for the constant. The outcome, \( 2\pi r \) is not just a number; it represents something tangible—the circumference of the circle itself—thus revealing an impactful connection between mathematical theory and concrete geometry.