Question

Evaluating a Derivative In Exercises \(65-72,\) find and evaluate the derivative of the function at the given point. Use a graphing utility to verify your result. $$ f(t)=\frac{3 t+2}{t-1}, \quad(0,-2) $$

Short answer

The derivative of the function \( f(t)=\frac{3 t+2}{t-1} \) is \( f'(t) = -5/(t-1)^2 \). Evaluating this at the point (0,-2), we get \( f'(0) = -5 \).

Step-by-step solution

Identify the functions u and v

In the given function, the numerator is \( u =3t+2 \) and the denominator is \( v = t-1 \). We need to find their derivatives u' and v' next.

Differentiate u and v

The derivative of \( u =3t+2 \) with respect to t is \( u' = 3 \). The derivative of \( v = t-1 \) with respect to t is \( v' = 1 \).

Apply the Quotient Rule

Applying the quotient rule, the derivative of \( f(t) \), \( f'(t) \), is given by \( (u'v - uv') / v^2 = (3(t-1) - (3t+2)(1))/ (t-1)^2 = (3t - 3 - 3t -2) / (t-1)^2 = -5/(t-1)^2 \).

Evaluate the Derivative at Given Point

We substitute \( t=0 \) into \( f'(t) = -5/(t-1)^2 \) to find the derivative at the given point (0, -2): \( f'(0) = -5/(-1)^2 = -5 \).

Verify Result

Use a graphing utility to plot the function and its derivative. Look at the slope of the tangent line at the point (0, -2) to confirm it equals to -5.

Key concepts

Quotient Rule

When you're faced with a function that is the ratio of two differentiable functions, such as \( f(t) = \frac{u(t)}{v(t)} \), the Quotient Rule comes into play to find the derivative of this compound function.

Here's a snapshot of how it works: You have two functions, \( u(t) \) and \( v(t) \) — think of them as the top and bottom parts of a fraction, respectively. To find the derivative \( f'(t) \) using the Quotient Rule, you'll calculate \( u'(t) \) and \( v'(t) \) first, which are the derivatives of \( u(t) \) and \( v(t) \) alone. Then, you put it all together: \( f'(t) = \frac{u'v - uv'}{v^2} \). That is, subtract the product of the derivative of the top function and the bottom function from the product of the top function and the derivative of the bottom function. Finally, divide the result by the square of the bottom function.

Derivative of a Function

The derivative is a measure that gives us the rate at which a function's output value is changing at any given point. One can think of it as an instantaneous rate of change, or the slope of the tangent line to the function's graph at a specific point.

Simple Derivatives

When the function is simple, such as a linear expression like \( u(t) = 3t + 2 \), its derivative, found by using basic derivative rules, is straightforward: \( u'(t) = 3 \).

Derivative at a Point

Moreover, evaluating the derivative at a specific point gives you the slope of the tangent at that particular point on the graph, telling you how steeply the function is rising or falling there. For instance, after applying the Quotient Rule, we evaluate the derivative at \( t = 0 \) to find the steepness of the tangent at that location in the graph.

Graphing Utilities

In mathematics, particularly in calculus, graphing utilities are valuable tools that provide visual contexts for understanding functions and their derivatives. These tools can range from simple graphing calculators to advanced computer software.

They allow students to plot the behavior of a function and its derivatives, making it easier to comprehend the concepts like the slope of a tangent line and the rate of change of a function. When you're vetted to verify your result, plotting \( f(t) \) and its derivative can visually confirm that the slope at the given point matches the value calculated using the derivative.

Tangent Line Slope

The slope of a tangent line to a graph at a given point represents the instantaneous rate of change of the function at that point. It's the slope that you get from the derivative of the function.

When you evaluate the derivative at a particular point, like \( t=0 \) in our exercise, you get the numerical value for this slope. In practical terms, if this slope is positive, the function is increasing at that point, and if it's negative, the function is decreasing. A slope of zero signifies a flat or a horizontal tangent, indicating a local maximum or minimum at that point.