The graph of 𝑦 equals negative one minus two sin 𝑥 has a zero on the closed interval 𝜋 by two to three 𝜋 by two. What is the slope of the tangent to the graph at that point?
Well firstly, we know that, to find the slope of the tangent to a graph when given a function 𝑓 of 𝑥, we need to evaluate the derivative at that point. The problem is, we don’t know where that point is yet. So we use the fact that the graph of 𝑦 equals negative one minus two sin of 𝑥 has a zero on the closed interval between 𝜋 by two and three 𝜋 by two radians. 𝜋 has a zero. This tells us that the graph intersects the 𝑥-axis in this interval.
And so to find the value of 𝑥 at this point, we let 𝑦 be equal to zero. So we get negative one minus two sin of 𝑥 equals zero. We need to solve for 𝑥. To achieve this, we add two sin 𝑥 to both sides of our equation and then divide by two, so negative one-half equals sin of 𝑥. Then we take the inverse or arcsin of both sides of our equation. So we find 𝑥 is equal to the inverse sin of negative one-half.
Well, we know this is true for one value; specifically, that’s 𝜋 by six. The problem is, that’s outside of our closed interval. However, we know that, for the sine curve, there is another solution. And we find that by subtracting 𝜋 by six from 𝜋. And we can find that value either by sketching the graph of 𝑦 is equal to sin 𝑥 and looking for various points of symmetry or by using the CAST diagram.
Now we need to find the slope of the tangent to the graph at the point where 𝑥 equals five 𝜋 by six. Well, the slope is found by differentiating 𝑦 with respect to 𝑥. We differentiate each term in turn. The derivative of negative one is zero. And when we differentiate negative two sin 𝑥, we get negative two cos 𝑥. Let’s evaluate this at the point where 𝑥 equals five 𝜋 by six. So that’s negative two times cos of five 𝜋 by six, which is equal to the square root of three.
And so the slope of the tangent to the graph of 𝑦 equals negative one minus two sin 𝑥 at the point where 𝑥 equals five 𝜋 by six is root three.