Q. 64 - Calculus

Question

Use the Intermediate Value Theorem to show that for each function f and value K in Exercises 61–66, there must be some c∈R for which f(c) = K. You will have to select an appropriate interval [a, b] to work with. Then find or approximate one such value of c. You may assume that these functions are continuous everywhere.

$$\begin{gathered} f\left(x\right)=\sin x; \\ K=\frac{\sqrt{3}}{2} \end{gathered}$$

Step-by-Step Solution

Answer

The approximate value of c of the function at $K=\frac{\sqrt{3}}{2}$ is $c\approx 1.05$.

Step 1. Given Information.

The function: 

$$\begin{gathered} f\left(x\right)=\sin x; \\ K=\frac{\sqrt{3}}{2} \end{gathered}$$

Step 2. Approximate the interval.

By trial and error we can find such values a and b, by testing different values of f(x) until we find one that is less than and one that is greater than $K=\frac{\sqrt{3}}{2}$.

$$\begin{gathered} f\left(\frac{\mathrm{\pi }}{4}\right)=\sin \frac{\mathrm{\pi }}{4} \\ =\frac{1}{\sqrt{2}}<\frac{\sqrt{3}}{2} \\ f\left(\frac{\mathrm{\pi }}{2}\right)=\sin \frac{\mathrm{\pi }}{2} \\ =1>\frac{\sqrt{3}}{2} \end{gathered}$$

Step 3. Apply intermediate value theorem.

Since f is continuous on $[\frac{\mathrm{\pi }}{4},\frac{\mathrm{\pi }}{2}]$ and $f\left(\frac{\mathrm{\pi }}{4}\right)<\frac{\sqrt{3}}{2}<f\left(\frac{\mathrm{\pi }}{2}\right)$, by the Intermediate Value Theorem there is some value $c$ for which $f\left(c\right)=\frac{\sqrt{3}}{2}$.

Note that the Intermediate Value Theorem doesn’t tell us where c is, only that such a c exists somewhere in the interval.

Step 4. Approximate c.

We can approximate some values of c for which $f\left(c\right)=\frac{\sqrt{3}}{2}$ by approximating the values
of x for which the graph of $f\left(x\right)=\sin x$ intersects the line $y=\frac{\sqrt{3}}{2}$.

From this graph we can conclude that $f\left(c\right)=\frac{\sqrt{3}}{2} at c\approx 1.05$