Problem 12 - Calculus

Question

Find the derivative of \(1 / t^{2}\) from \(\Delta f(t)=1 /(t+\Delta t)^{2}-1 / t^{2}\) Write \(\Delta f\) as a fraction with the denominator \(t^{2}(t+\Delta t)^{2}\). Divide the numerator by \(\Delta t\) to find \(\Delta f / \Delta t\). Set \(\Delta t=0\).

Step-by-Step Solution

Answer

The derivative of \(\frac{1}{t^2}\) is \(\frac{-2}{t^3}\).

Step 1: Rewrite Function in Terms of Fraction

We start by rewriting \(\Delta f(t)\) as a single fraction. Given:\[\Delta f(t) = \frac{1}{(t+\Delta t)^2} - \frac{1}{t^2}\]We express it with a common denominator \(t^2(t+\Delta t)^2\), which is the product of the two individual denominators. This gives us:\[\Delta f(t) = \frac{t^2 - (t + \Delta t)^2}{t^2(t+\Delta t)^2}\]

Step 2: Expand and Simplify the Numerator

Next, we expand the terms in the numerator:\[\Delta f(t) = \frac{t^2 - (t^2 + 2t\Delta t + (\Delta t)^2)}{t^2(t+\Delta t)^2}\]Simplifying the expression, we get:\[\Delta f(t) = \frac{t^2 - t^2 - 2t\Delta t - (\Delta t)^2}{t^2(t+\Delta t)^2}\]This reduces to:\[\Delta f(t) = \frac{-2t\Delta t - (\Delta t)^2}{t^2(t+\Delta t)^2}\]

Step 3: Factor Out \(\Delta t\) from the Numerator

We can factor \(\Delta t\) from the numerator to prepare for dividing by \(\Delta t\):\[\Delta f(t) = \frac{-\Delta t(2t + \Delta t)}{t^2(t+\Delta t)^2}\]

Step 4: Divide by \(\Delta t\) to Find \(\frac{\Delta f}{\Delta t}\)

Now, divide the entire fraction by \(\Delta t\):\[\frac{\Delta f}{\Delta t} = \frac{-\Delta t(2t + \Delta t)}{t^2(t+\Delta t)^2} \div \Delta t = \frac{-(2t + \Delta t)}{t^2(t+\Delta t)^2}\]

Step 5: Set \(\Delta t = 0\) and Evaluate the Limit

Finally, we evaluate the expression as \(\Delta t\) approaches \(0\):\[\frac{-(2t + \Delta t)}{t^2(t+\Delta t)^2} \xrightarrow{\Delta t \to 0} \frac{-2t}{t^2(t)^2} = \frac{-2t}{t^4} = \frac{-2}{t^3}\]Thus, the derivative of \(\frac{1}{t^2}\) is \(\frac{-2}{t^3}\).

Key Concepts

Limits

The concept of limits is foundational in calculus and plays a crucial role when finding derivatives. A limit essentially helps us determine what value a function approaches as the input gets closer to a certain point. In the context of finding a derivative, we often use limits to assess the behavior of a function as the change in the input variable \(\Delta t\) approaches zero.

When calculating the derivative of \(\frac{1}{t^2}\), we took \(\Delta f(t)\) and expressed it as a difference quotient - \(\frac{\Delta f}{\Delta t}\).- The limit is applied to this quotient. As \(\Delta t\) tends to zero, the expression simplifies, providing the instantaneous rate of change of the function, which is the derivative. This step was done in the solution by first expressing the difference quotient and then letting \(\Delta t \to 0\).

In simple terms, by using limits, we can find how fast \(\frac{1}{t^2}\) changes with respect to \t\ at any given point, resulting in its derivative, \(\frac{-2}{t^3}\).

• Limits allow calculus to deal with changes smoothly.
• The limit value helps us traverse from a secant line (average change) to the tangent line (instantaneous change).

Fraction Simplification

Simplifying fractions is an essential skill, particularly when dealing with complex algebraic expressions in calculus. In the original solution, we started with \(\Delta f(t) = \frac{1}{(t+\Delta t)^2} - \frac{1}{t^2}\). The task was to combine these into a single fraction. This involves finding a common denominator, which in this case was \(t^2(t+\Delta t)^2\).

After finding the common denominator, the next step was to combine the fractions into a single fraction by expressing each term over the common denominator, resulting in:

• \(\Delta f(t) = \frac{t^2 - (t + \Delta t)^2}{t^2(t+\Delta t)^2}\)

This expression was further simplified by expanding and rearranging the numerator, allowing for easier manipulation when calculating the derivative. Simplification like this not only makes the expressions more manageable but also prepares them for subsequent operations, such as taking limits.

When fractions are simplified, calculation errors can be reduced, and expressions are easier to understand and compute. Always remember to carry out simplifications step-by-step, clearly showing each stage as was done in the solution.

Algebraic Manipulation

Algebraic manipulation involves changing and re-arranging expressions to simplify calculations. In calculus, especially when finding derivatives, it is often necessary to manage complex expressions for clarity and simplicity. This includes expanding, factoring, and simplifying fractions, as seen in the solution.

For instance, you saw how the expression \(t^2 - (t^2 + 2t\Delta t + (\Delta t)^2)\) was expanded and simplified to lead to \(-2t\Delta t - (\Delta t)^2\). Factors were then manipulated to isolate \(\Delta t\) making it easier to cancel out later. Factoring out common terms, like \(\Delta t\), is crucial since it leads the way to dividing the numerator by \(\Delta t\) and eventually applying limits to find the derivative.

These manipulations are what connect algebra to calculus, allowing us to build bridges from average rates of change (secant lines) to instantaneous rates of change (tangent lines).

• Algebraic manipulation helps simplify expressions.
• This simplification is essential for applying limits and finding derivatives efficiently.
• Skills in algebra can greatly ease calculus operations and understanding.