Problem 10 - Single Variable Calculus

Question

In Exercises \(7-24\), sketch the graph of the function and find its absolute maximum and absolute minimum values, if any. $$ f(t)=t^{2}-1 \text { on }[-1,0) $$

Step-by-Step Solution

Answer

The graph of the function \(f(t)=t^2-1\) on the interval [-1,0) is a parabola opening upwards, with its vertex as t approaches 0 from the left and f(t)=-1. On this interval, the function has an absolute minimum value of \(-1\) as \(t \to 0^{-}\), and an absolute maximum value of \(0\) at \(t = -1\).

Step 1: Find the critical points

To find the critical points of \(f(t) = t^2 - 1\), we first need to determine its derivative. The derivative of the function is: \(f'(t) = \frac{d}{dt}(t^2 - 1) = 2t\) A critical point occurs when the derivative is equal to zero or is undefined. Since the derivative is never undefined for this function, we just need to set it equal to zero and solve for t: \(2t = 0\) \(t = 0\) However, since our interval is [-1, 0), and 0 is not included in the interval, we don't have any critical points in the given interval.

Step 2: Evaluate the function at the endpoints

Now, we will evaluate the function at the endpoints of the given interval, -1 and 0 (not included, so we will consider its limit): \(f(-1) = (-1)^2 -1 = 0\) Since 0 is not included in the interval, we will now find the limit of the function as t approaches 0 from the left side: \(\lim_{t \to 0^{-}} (t^2 - 1) = (0)^2 - 1 = -1\)

Step 3: Determine the maximum and minimum values

From the values obtained in the previous step, we can now determine the maximum and minimum values of the function on the given interval: The absolute minimum value is the lowest value, which is \(f(t) = -1\), and it occurs when \(t \to 0^{-}\). The absolute maximum value is the highest value, which is \(f(t) = 0\), and it occurs at \(t = -1\).

Step 4: Sketch the graph of the function

To sketch the graph of the function, plot the function \(f(t) = t^2 - 1\) on the interval [-1, 0). The graph will be a parabola opening upwards, with its vertex at t=0 and f(t)=-1. Since the parabola opens upwards, it will start from the endpoint at t=-1 and f(t)=0, reach its lowest point (vertex) as t approaches 0 from the left side, and f(t)=-1, and then open upwards (but this part will not be included in our sketch as our interval is only up to t=0).

Key Concepts

Critical Points in Calculus

In calculus, critical points are crucial for understanding the behavior of functions. They are the points on the graph where the function's derivative is equal to zero or where the derivative does not exist. Critical points can signal where a function reaches a local maximum or minimum, or where the slope of the tangent line is horizontal.

To find the critical points, one must first take the derivative of the function. For the function in our exercise, the first derivative was found to be \( f'(t) = 2t \). The next step is to set this derivative equal to zero and solve for \( t \). In this case, we find that the critical point is at \( t = 0 \), but this point is not included in the interval \( [-1, 0) \).

This means that for this particular exercise, we do not have any critical points within the given interval, but in a more general problem, once the critical points are found, they are used in conjunction with the values of the function at the endpoints of the interval to determine relative maximums, minimums, and to assess the overall shape and turning points of the graph.

Function Intervals

Understanding function intervals is essential when sketching graphs and determining the behavior of functions. An interval represents a range of values along the domain of the function, usually between two points.

In this exercise, the interval given is \( [-1, 0) \), which includes \( -1 \) but not \( 0 \). The interval notation indicates the parts of the graph we're interested in evaluating. It is important to consider the endpoints of the interval, as they can provide information about absolute minimum and maximum values and also help define the extent of the graph.

For instance, when \( f(t) = t^2 - 1 \) is evaluated at \( t = -1 \), it yields an output of \( 0 \). As \( t \) approaches the other endpoint \( 0 \) from the left, the limit of the function must be considered, since \( 0 \) is not included in the interval. By evaluating these points, we gain a clearer understanding of the section of the graph we are interested in and how the function behaves on this specific interval.

Absolute Maximum and Minimum Values

The concepts of absolute maximum and minimum values are essential when analyzing the complete behavior of a function over a closed interval. An absolute maximum is the highest value that the function reaches on the interval, while an absolute minimum is the lowest.

In this particular exercise, after evaluating the endpoints of the interval and considering the function's limit as \( t \) approaches the non-inclusive endpoint, we find the absolute maximum and minimum values. The absolute minimum value of \( f(t) \) on the interval \( [-1, 0) \) turns out to be \( -1 \), occurring as \( t \) approaches \( 0^- \). Conversely, the absolute maximum value is \( 0 \) which is found at \( t = -1 \).

Knowing these values is critical for graph sketching because they represent the highest and lowest points we will plot, defining the range of our function on the interval specified. In graphing exercises, like the one at hand, highlighting these values helps visualize the function's growth and decay, providing an insight into its overall behavior within a given domain.