Problem 18 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(15-18\) , determine whether the curve has a tangent at the indicated point, If it does, give its slope, If not, explain why not. $$f(x)=\left\\{\begin{array}{ll}{\sin x,} & {0 \leq x<3 \pi / 4} \\ {\cos x,} & {3 \pi / 4 \leq x \leq 2 \pi}\end{array}\right.\( at \)x=3 \pi / 4$$

Step-by-Step Solution

Answer

The curve \( f(x) \) does have a tangent line at \( x = 3 \pi / 4 \) and its slope is \(-\sqrt{2}/2\).

Step 1: Find the derivative of \( \sin x \) and \( \cos x \)

Derivate separately the two parts of the function:The derivative \(\sin x\) is \(\cos x\), and the derivative of \(\cos x\) is \(-\sin x\)

Step 2: Evaluate these derivatives at \( x = 3 \pi / 4 \)

To do this, we will substitute \( 3 \pi / 4 \) into \(\cos x\), which gives \(\cos(3 \pi / 4)\). This yields \(-\sqrt{2}/2\).Substituting \( 3 \pi / 4 \) into \(-\sin x\), gives \(-\sin(3 \pi / 4)\). This yields \(-\sqrt{2}/2\).

Step 3: Compare the two results

Seeing that both limits are equal, we can conclude that the limit as \( x \) approaches \( 3 \pi / 4 \) of the derivative exists and therefore there is a tangent at \( x = 3 \pi / 4 \)

Key Concepts

Derivatives

In calculus, derivatives are a fundamental concept that measure how a function changes as its input changes. Consider them as the function's instantaneous rate of change at any given point.

• For a function such as \(f(x) = \sin x\), the derivative, \(f'(x)\), tells us the slope of the curve at any point \(x\).
• The derivative of \(\sin x\) is \(\cos x\).
• For \(f(x) = \cos x\), the derivative is \(-\sin x\).

Using these derivatives, you can calculate the slope of a curve at different points. For instance, finding the derivative at a specific point like \(x = 3 \pi / 4\) helps us understand how the function behaves around that value. To derive composite functions, like piecewise functions, we derive each piece separately and carefully consider the points where these functions switch.

Piecewise Functions

Piecewise functions are defined by different expressions for different intervals of the domain. They let us model scenarios where a rule might change based on input. In our exercise, the function \(f(x)\) is defined as follows:

• \(f(x) = \sin x\) for \(0 \leq x < 3\pi/4\)
• \(f(x) = \cos x\) for \(3\pi/4 \leq x \leq 2\pi\)

Working with piecewise functions requires special attention at the transition points, like \(x = 3 \pi / 4\). At these specific points, it’s essential to ensure that derivatives match to maintain continuity and differentiability.To determine if there is a tangent at a transition point, we analyze the derivatives from both sides.

Tangent Slope

Finding the tangent slope at a specific point on a curve involves calculating the derivative at that point. The tangent line is the line that just 'touches' the curve at the point, having the same slope as the curve right there.For a piecewise function, if both parts have the same derivative at the transition point, the function has a tangent there:

• For \( f(x) = \sin x \), the derivative at \( x = 3 \pi / 4 \) is \( \cos(3 \pi / 4) = -\sqrt{2}/2 \).
• For \( f(x) = \cos x \), the derivative at \( x = 3 \pi / 4 \) is \( -\sin(3 \pi / 4) = -\sqrt{2}/2 \).

Since both derivatives are equal, a tangent exists at \( x = 3 \pi / 4 \), and it has a slope of \( -\sqrt{2}/2 \). A consistent tangent suggests the function smoothly passes through this point without a sharp corner or cusp.