Problem 3 - Calculus

Question

In Problems \(I-10\), use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. $$ h(t)=t^{2}+2 t-3 $$

Step-by-Step Solution

Answer

The function is decreasing on \((-\infty, -1)\) and increasing on \((-1, \infty)\).

Step 1: Understanding the Monotonicity Theorem

To determine where a function is increasing or decreasing, we first need to find its derivative. A function \( f(x) \) is increasing on an interval if its derivative \( f'(x) > 0 \) for every point in that interval; it is decreasing if \( f'(x) < 0 \) for every point in the interval.

Step 2: Finding the Derivative

Calculate the derivative of the given function. The function is \( h(t) = t^{2} + 2t - 3 \). Using the power rule: \( h'(t) = 2t + 2 \).

Step 3: Setting the Derivative Equal to Zero

To find critical points, where the direction of the function can change, set \( h'(t) = 0 \).\[ 2t + 2 = 0 \] \[ 2t = -2 \] \[ t = -1 \]

Step 4: Determine Intervals for Testing

Use \( t = -1 \) to divide the real line into intervals. The intervals are: \(( -\infty, -1 ) \) and \( (-1, \infty) \).

Step 5: Testing Intervals

Test a point in each interval to determine if the function is increasing or decreasing:- For \( t = -2 \) (in \( ( -\infty, -1 ) \)): \( h'(-2) = 2(-2) + 2 = -4 + 2 = -2 \) (negative, so decreasing)- For \( t = 0 \) (in \( (-1, \infty) \)): \( h'(0) = 2(0) + 2 = 2 \) (positive, so increasing)

Step 6: Conclusion

Based on the sign of the derivative in each interval, we conclude:- The function \( h(t) \) is decreasing on \( (-\infty, -1) \).- The function \( h(t) \) is increasing on \( (-1, \infty) \).

Key Concepts

Monotonicity Theorem

The Monotonicity Theorem is a fundamental concept in calculus that helps us determine where a function is increasing or decreasing. This theorem states that a function is increasing on an interval if its derivative is positive throughout the interval. Conversely, the function is decreasing if its derivative is negative. The key idea here is the relationship between the derivative and the behavior of the function. For any given function, by studying its derivative, you can predict how the function behaves across its domain. Using the derivative, you can also find critical points, where the derivative is zero or undefined, which indicate potential changes in the increasing or decreasing nature of the function. Critical points help divide the function into clear sections for deeper analysis.

Derivative

The derivative of a function is a core component in calculus, quantifying how the function value changes concerning its input. It is the mathematical representation of the rate of change or slope of a function at any given point. For a function expressed as \( h(t) = t^{2} + 2t - 3 \), calculating the derivative involves applying differentiation rules. Using the power and constant rules, the derivative, \( h'(t) = 2t + 2 \), reflects the slope of \( h(t) \) at any value of \( t \).

- *Power Rule*: The derivative of \( t^n \) is \( nt^{n-1} \), where \( n \) is a real number.
- *Constant Rule*: The derivative of any constant is 0.
Differentiating correctly allows you to evaluate the function's slope to make predictions about its behavior, such as where the function might be increasing or decreasing.

Increasing and Decreasing Functions

Understanding where a function increases or decreases is crucial for identifying its overall behavior and attributes like peaks and troughs. After differentiating the function, as shown in the case of \( h(t) = t^{2} + 2t - 3 \), the derivative \( h'(t) \) is used to find intervals over which the function either ascends or descends.

- **Increasing**: When the derivative \( h'(t) \) is greater than 0, the function is rising over that interval. For \( h(t) \), it increases on the interval \((-1, \infty)\).
- **Decreasing**: If the derivative is less than 0, the function is declining. In this case, \( h(t) \) decreases over the interval \((-\infty, -1)\).
By testing points within each interval, you can validate whether the function is indeed increasing or decreasing. This knowledge is vital in various fields like physics, economics, and any domain where it’s important to understand how changes in variables affect outcomes over time.