Calculus Test 2 Review: Step-by-Step Guidance for Key Questions

Q2. Graph the curve y = cos(x) over the interval $-2\pi \leq x \leq 2\pi$, and graph its tangent at $x = \frac{\pi}{2}$ with a dashed line.

Background

Topic: Graphing Functions and Tangent Lines

This question tests your ability to graph a trigonometric function and to find and graph the tangent line at a specific point. Understanding how to compute derivatives and interpret them geometrically is essential.

Key Terms and Formulas:

• Cosine Function: $y = \cos(x)$

• Derivative: $y' = -\sin(x)$

• Tangent Line Equation: $y = f'(a)(x - a) + f(a)$, where $a$ is the point of tangency.

Step-by-Step Guidance

1. Sketch the graph of $y = \cos(x)$ over the interval $-2\pi \leq x \leq 2\pi$. Note the periodic nature and amplitude of the cosine function.

2. Find the derivative of $y = \cos(x)$, which is $y' = -\sin(x)$. This gives the slope of the tangent at any point $x$.

3. Evaluate the slope at $x = \frac{\pi}{2}$: $y'\left(\frac{\pi}{2}\right) = -\sin\left(\frac{\pi}{2}\right)$.

4. Find the value of $y$ at $x = \frac{\pi}{2}$: $y\left(\frac{\pi}{2}\right) = \cos\left(\frac{\pi}{2}\right)$.

5. Set up the equation for the tangent line using the point-slope form: $y = y'\left(\frac{\pi}{2}\right)(x - \frac{\pi}{2}) + y\left(\frac{\pi}{2}\right)$.

Try solving on your own before revealing the answer!

Final Answer:

The tangent line at $x = \frac{\pi}{2}$ has slope $-1$ and passes through the point $(\frac{\pi}{2}, 0)$. The equation is $y = -1 \cdot (x - \frac{\pi}{2})$.

On the graph, the tangent line is drawn as a dashed line at $x = \frac{\pi}{2}$, showing the instantaneous rate of change of $y = \cos(x)$ at that point.