Problem 8
Question
Determine the values of \(c\) at which \(f^{\prime}\) changes from positive to negative, or from negative to positive. $$ f(t)=\sin t-\cos t $$
Step-by-Step Solution
Verified Answer
The values of \( c \) are \( \frac{3\pi}{4} + n\pi \).
1Step 1: Find the Derivative
Given the function \( f(t) = \sin t - \cos t \), we find its derivative: \( f'(t) = \cos t + \sin t \). This is derived by differentiating \( \sin t \) and \( -\cos t \) individually.
2Step 2: Set the Derivative Equal to Zero
Determine where \( f'(t) = \cos t + \sin t = 0 \). These points will help us find where the function's slope changes from positive to negative, or vice versa.
3Step 3: Solve for \( t \)
Simplify \( \cos t + \sin t = 0 \) to \( \cos t = -\sin t \). Dividing both sides by \( \cos t \) (where \( \cos t eq 0 \)), we get \( \tan t = -1 \). Thus, \( t = \frac{3\pi}{4} + n\pi \) for any integer \( n \).
4Step 4: Test for Sign Change
Check intervals around \( t = \frac{3\pi}{4} + n\pi \) to determine where \( f'(t) \) changes sign. You can test values slightly before and after a point, such as \( t = \frac{3\pi}{4} \) or \( t = \frac{7\pi}{4} \), to see if \( f'(t) \) changes from positive to negative or vice versa.
Key Concepts
DerivativeTrigonometric FunctionsSign ChangeSlope Analysis
Derivative
The concept of a derivative is central in calculus. It represents the rate at which a function changes at any given point. For a function like \( f(t) = \sin t - \cos t \), finding the derivative means calculating how the function's value changes as \( t \) changes slightly.
To find the derivative \( f'(t) \), differentiate each term in the function separately. The derivative of \( \sin t \) is \( \cos t \), and for \( -\cos t \), it is \( \sin t \). Thus, the derivative becomes \( f'(t) = \cos t + \sin t \). This equation will be the tool we use to locate points where the slope of the original function changes.
To find the derivative \( f'(t) \), differentiate each term in the function separately. The derivative of \( \sin t \) is \( \cos t \), and for \( -\cos t \), it is \( \sin t \). Thus, the derivative becomes \( f'(t) = \cos t + \sin t \). This equation will be the tool we use to locate points where the slope of the original function changes.
Trigonometric Functions
Trigonometric functions such as \( \sin t \) and \( \cos t \) frequently appear in calculus problems. They are periodic functions, meaning they repeat their values over regular intervals (specifically \(2\pi\) radians for sine and cosine).
When you differentiate these functions, their interrelated nature shines. For instance, \( \sin t \) becomes \( \cos t \) upon differentiation. Conversely, when differentiating \(-\cos t\), it results in \( \sin t \). These transformations are crucial when analyzing the behavior of combined trigonometric functions like \( \sin t - \cos t \). The interdependence of these functions plays a key role in determining where the function's slope changes direction.
When you differentiate these functions, their interrelated nature shines. For instance, \( \sin t \) becomes \( \cos t \) upon differentiation. Conversely, when differentiating \(-\cos t\), it results in \( \sin t \). These transformations are crucial when analyzing the behavior of combined trigonometric functions like \( \sin t - \cos t \). The interdependence of these functions plays a key role in determining where the function's slope changes direction.
- Periodic nature: functions repeat every \(2\pi\).
- Derivatives switch sine to cosine and vice versa.
- Useful for analyzing function behavior over intervals.
Sign Change
A sign change in the derivative of a function indicates a change in the direction of the function's slope. For the function \( f(t) = \sin t - \cos t \), we look at the derivative \( f'(t) = \cos t + \sin t \).
The task is to determine when this derivative changes from positive to negative, or from negative to positive. This means solving for \( t \) when \( \cos t + \sin t = 0 \). Among the solutions, \( t = \frac{3\pi}{4} + n\pi \) (where \( n \) is an integer) tells us potential points of sign change. Testing values just before and after these points helps confirm the nature of the sign change. Discovering such intervals allows us to identify critical points where the function's slope behavior changes.
The task is to determine when this derivative changes from positive to negative, or from negative to positive. This means solving for \( t \) when \( \cos t + \sin t = 0 \). Among the solutions, \( t = \frac{3\pi}{4} + n\pi \) (where \( n \) is an integer) tells us potential points of sign change. Testing values just before and after these points helps confirm the nature of the sign change. Discovering such intervals allows us to identify critical points where the function's slope behavior changes.
Slope Analysis
Slope analysis comes into play by examining the points where the derivative changes sign. From calculus, we know a positive derivative means the function is increasing, and a negative derivative indicates the function decreases.
For \( f'(t) = \cos t + \sin t \), testing intervals around \( t = \frac{3\pi}{4} + n\pi \) will reveal the slope change. For example, plugging in values slightly less and more than \( t = \frac{3\pi}{4} \) into \( f'(t) \) shows whether the slope goes from positive to negative or negative to positive.
For \( f'(t) = \cos t + \sin t \), testing intervals around \( t = \frac{3\pi}{4} + n\pi \) will reveal the slope change. For example, plugging in values slightly less and more than \( t = \frac{3\pi}{4} \) into \( f'(t) \) shows whether the slope goes from positive to negative or negative to positive.
- Positive slope: function increasing.
- Negative slope: function decreasing.
- Critical points determined by testing vicinity of solutions.
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