Problem 20 - Calculus and its applications

Question

Differentiate two ways: first, by using the Quotient Rule; then, by dividing the expressions before differentiating. Compare your results as a check. Use a graphing calculator to check your results. \(y=\frac{t^{2}-25}{t-5}\)

Step-by-Step Solution

Answer

Both methods confirm the derivative is 1 for \( t \neq 5 \).

Step 1: Apply the Quotient Rule

To differentiate the function \( y = \frac{t^2 - 25}{t - 5} \) using the Quotient Rule, identify the numerator as \( u = t^2 - 25 \) and the denominator as \( v = t - 5 \). The Quotient Rule states: \( \frac{d}{dt} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2} \).Differentiate: \( u' = 2t \), \( v' = 1 \).Substitute back into the Quotient Rule: \[ \frac{d}{dt} \left( \frac{t^2 - 25}{t - 5} \right) = \frac{(2t)(t - 5) - (t^2 - 25)(1)}{(t - 5)^2} \].Simplify the expression: the numerator becomes \( 2t^2 - 10t - t^2 + 25 = t^2 - 10t + 25 \). Thus, the derivative is: \[ \frac{d}{dt} \left( \frac{t^2 - 25}{t - 5} \right) = \frac{t^2 - 10t + 25}{(t - 5)^2} \].

Step 2: Simplify Before Differentiating

Now, simplify the function \( y = \frac{t^2 - 25}{t - 5} \) before differentiating. Notice that \( t^2 - 25 \) can be factored as \( (t - 5)(t + 5) \), so we have:\[ y = \frac{(t - 5)(t + 5)}{t - 5} = t + 5 \] assuming \( t eq 5 \).Differentiate the simplified function directly: \[ \frac{d}{dt}(t + 5) = 1 \].

Step 3: Compare the Results

The differentiated function using the Quotient Rule is \( \frac{t^2 - 10t + 25}{(t - 5)^2} \), while simplifying first gives \( 1 \).Notice that the steps will match when you eliminate common terms and then substitute specific values of \( t \) other than \( t = 5 \) in both expressions to confirm the correctness.

Step 4: Verify with Graphing Calculator

Use a graphing calculator to graph both \( \frac{t^2 - 25}{t - 5} \) and \( t + 5 \) to see that they are equivalent for \( t eq 5 \). Check their derivatives with respect to \( t \) at points other than \( t = 5 \) to confirm that both indicate the derivative as \( 1 \).

Key Concepts

Differentiation

The process of differentiation involves finding the derivative of a function, which tells us the rate at which the function's value changes with respect to changes in its input. In this exercise, we begin with the function \( y = \frac{t^2 - 25}{t - 5} \). To differentiate this, we use the Quotient Rule. The Quotient Rule is helpful for functions that are ratios, i.e., one function divided by another.

Here's a quick recap of the rule: For a function \( y = \frac{u}{v} \), the derivative is given by

• \( y' = \frac{u'v - uv'}{v^2} \)

where \( u \) is the numerator and \( v \) is the denominator. Each of these components must be separately differentiated.

This exercise outcome with the Quotient Rule shows that we get a more involved expression, which simplifies differentiation where necessary and checks the accuracy of derivative calculations. However, always check if further simplification of the function can provide an easier path for differentiation.

Simplification

Simplification before differentiation can sometimes lead to a much easier process, as seen in the exercise example with \( y = \frac{t^2 - 25}{t - 5} \).

Before diving into differentiation using complex rules, check if the expression can be simplified:

• The term \( t^2 - 25 \) is recognized as a difference of squares, which can be rewritten as \((t - 5)(t + 5)\).
• This simplification allows cancellation with the denominator \( t - 5 \), resulting in the expression \( y = t + 5 \).

Differentiating \( y = t + 5 \) is straightforward:

• The derivative is simply \( \frac{d}{dt}(t + 5) = 1 \).

Such simplification not only makes the problem easier but also significantly reduces the complexity of calculations—saving time and reducing potential errors. Always consider simplifying expressions where possible before applying differentiation rules.

Graphing Calculator

To verify the results obtained through differentiation, a graphing calculator can be extremely useful. A graphing calculator can graph functions and their derivatives, offering a visual check on the calculations.

To use a graphing calculator:

• Graph both the original function \( \frac{t^2 - 25}{t - 5} \) and the simplified version \( t + 5 \) to observe that they produce the same graph for \( t eq 5 \).
• Calculate derivatives at specific points other than \( t = 5 \) using graphing tools to ensure that derivative values match (both yielding a derivative of 1).

Having a clear visual confirmation through the graph helps ensure your algebraic steps were executed correctly. Graphing calculators serve as an excellent check and can foster deeper understanding by providing concrete visualizations.