Math  /  Trigonometry

QuestionFor help with questions 1 and 2, refer to the Investigate.
1. a) Sketch a graph of f(x)=cosxf(x)=\cos x on the interval x[0,2π]x \in[0,2 \pi]. b) For what values of xx does the instantaneous rate of change appear to equal 0 ? c) For what values of xx does the instantaneous rate of change appear to reach a maximum value? a minimum value?

Studdy Solution

STEP 1

What is this asking? We're going to draw the cosine graph and see where its slope is zero, biggest, and smallest! Watch out! Remember, instantaneous rate of change means the *slope* at a single point.

STEP 2

1. Sketch the graph
2. Find zero slopes
3. Find max/min slopes

STEP 3

Alright, let's **sketch** cos(x)\cos(x) between 00 and 2π2\pi.
Remember, cosine starts at its **peak**!
So at x=0x = 0, cos(x)=1\cos(x) = 1.

STEP 4

Then, it smoothly goes down to zero at x=π2x = \frac{\pi}{2}, hits its **lowest point** at x=πx = \pi where cos(x)=1\cos(x) = -1, back to zero at x=3π2x = \frac{3\pi}{2}, and finally returns to its **peak** at x=2πx = 2\pi where cos(x)=1\cos(x) = 1 again!

STEP 5

Now, where is the slope zero?
Imagine a little car driving along the curve.
Where would it be perfectly flat?

STEP 6

That happens at the **peaks** and **valleys**!
So the slope is zero at x=π2x = \frac{\pi}{2}, x=3π2x = \frac{3\pi}{2}, and also at the endpoints x=0x=0 and x=2πx=2\pi.

STEP 7

Okay, where is the slope the steepest, meaning where does the car go downhill the fastest?
That happens at x=πx = \pi, right at the **bottom of the curve**!
The slope there is the most negative, so that's our **minimum** slope.

STEP 8

And where is the slope steepest uphill?
That's at the very beginning, x=0x=0, and the very end, x=2πx=2\pi!
The slope at these points is the most positive, so that's our **maximum** slope.

STEP 9

a) The sketch of cos(x)\cos(x) shows a wave-like curve starting at 11, going down to 1-1, and back up to 11. b) The instantaneous rate of change (slope) is zero at x=0x = 0, x=π2x = \frac{\pi}{2}, x=3π2x = \frac{3\pi}{2}, and x=2πx = 2\pi. c) The maximum instantaneous rate of change occurs at x=0x = 0 and x=2πx = 2\pi, and the minimum instantaneous rate of change occurs at x=πx = \pi.

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