Math / Calculus

QuestionLet $g(y)=7 \sin (y)$ Determine the average value, $g(c)$ , of $g$ over $\left[0, \frac{3 \pi}{4}\right]$ . $g(c)=$ $\square$ Determine the value(s) of $c$ in $\left[0, \frac{3 \pi}{4}\right]$ guaranteed by the Mean Value Theorem. Round the solution(s) to four decimal places, if necessary. $c=$ $\square$

Studdy Solution

STEP 1

1. The function $g(y) = 7 \sin(y)$ is continuous on the interval $\left[0, \frac{3\pi}{4}\right]$ .
2. The average value of a continuous function $f(x)$ over an interval $[a, b]$ is given by the formula: \frac{1}{b-a} \int_a^b f(x) \, dx \]
3. The Mean Value Theorem for Integrals states that if $ f $ is continuous on $[a, b]$, then there exists at least one $ c $ in $(a, b)$ such that: f(c) = \frac{1}{b-a} \int_a^b f(x) \, dx \]

STEP 2

1. Calculate the average value of $g(y)$ over $\left[0, \frac{3\pi}{4}\right]$ .
2. Determine the value(s) of $c$ in $\left[0, \frac{3\pi}{4}\right]$ that satisfy the Mean Value Theorem.

STEP 3

Calculate the average value of $g(y) = 7 \sin(y)$ over $\left[0, \frac{3\pi}{4}\right]$ :
The average value $g(c)$ is given by: $g(c) = \frac{1}{\frac{3\pi}{4} - 0} \int_0^{\frac{3\pi}{4}} 7 \sin(y) \, dy$

STEP 4

Evaluate the integral: $\int 7 \sin(y) \, dy = -7 \cos(y) + C$
Thus, $\int_0^{\frac{3\pi}{4}} 7 \sin(y) \, dy = \left[-7 \cos(y)\right]_0^{\frac{3\pi}{4}}$

STEP 5

Calculate the definite integral: $= -7 \cos\left(\frac{3\pi}{4}\right) + 7 \cos(0)$
Since $\cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}$ and $\cos(0) = 1$ , we have: $= -7 \left(-\frac{\sqrt{2}}{2}\right) + 7 \cdot 1$ $= \frac{7\sqrt{2}}{2} + 7$

STEP 6

Calculate the average value: $g(c) = \frac{1}{\frac{3\pi}{4}} \left(\frac{7\sqrt{2}}{2} + 7\right)$
Simplify: $g(c) = \frac{4}{3\pi} \left(\frac{7\sqrt{2}}{2} + 7\right)$

STEP 7

Determine the value(s) of $c$ in $\left[0, \frac{3\pi}{4}\right]$ such that: $g(c) = 7 \sin(c) = \frac{4}{3\pi} \left(\frac{7\sqrt{2}}{2} + 7\right)$

STEP 8

Solve for $c$ : $\sin(c) = \frac{4}{3\pi} \left(\frac{\sqrt{2}}{2} + 1\right)$
Calculate the numerical value: $\sin(c) \approx \frac{4}{3\pi} \left(\frac{1.414}{2} + 1\right)$

STEP 9

Calculate the approximate value of $c$ using the inverse sine function: $c = \arcsin\left(\frac{4}{3\pi} \left(\frac{\sqrt{2}}{2} + 1\right)\right)$
Compute $c$ to four decimal places.
The average value $g(c)$ is approximately $\boxed{\frac{4}{3\pi} \left(\frac{7\sqrt{2}}{2} + 7\right)}$ .
The value of $c$ is approximately $\boxed{\arcsin\left(\frac{4}{3\pi} \left(\frac{\sqrt{2}}{2} + 1\right)\right)}$ .

Was this helpful?

Studdy Solution

STEP 1

STEP 2

1. Calculate the average value of $g(y)$ over $\left[0, \frac{3\pi}{4}\right]$ .
2. Determine the value(s) of $c$ in $\left[0, \frac{3\pi}{4}\right]$ that satisfy the Mean Value Theorem.

STEP 3

Calculate the average value of $g(y) = 7 \sin(y)$ over $\left[0, \frac{3\pi}{4}\right]$ :
The average value $g(c)$ is given by: $g(c) = \frac{1}{\frac{3\pi}{4} - 0} \int_0^{\frac{3\pi}{4}} 7 \sin(y) \, dy$

STEP 4

Evaluate the integral: $\int 7 \sin(y) \, dy = -7 \cos(y) + C$
Thus, $\int_0^{\frac{3\pi}{4}} 7 \sin(y) \, dy = \left[-7 \cos(y)\right]_0^{\frac{3\pi}{4}}$

STEP 5

Calculate the definite integral: $= -7 \cos\left(\frac{3\pi}{4}\right) + 7 \cos(0)$
Since $\cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}$ and $\cos(0) = 1$ , we have: $= -7 \left(-\frac{\sqrt{2}}{2}\right) + 7 \cdot 1$ $= \frac{7\sqrt{2}}{2} + 7$

STEP 6

Calculate the average value: $g(c) = \frac{1}{\frac{3\pi}{4}} \left(\frac{7\sqrt{2}}{2} + 7\right)$
Simplify: $g(c) = \frac{4}{3\pi} \left(\frac{7\sqrt{2}}{2} + 7\right)$

STEP 7

Determine the value(s) of $c$ in $\left[0, \frac{3\pi}{4}\right]$ such that: $g(c) = 7 \sin(c) = \frac{4}{3\pi} \left(\frac{7\sqrt{2}}{2} + 7\right)$

STEP 8

Solve for $c$ : $\sin(c) = \frac{4}{3\pi} \left(\frac{\sqrt{2}}{2} + 1\right)$
Calculate the numerical value: $\sin(c) \approx \frac{4}{3\pi} \left(\frac{1.414}{2} + 1\right)$

STEP 9

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

STEP 2

1. Calculate the average value of g(y) g(y) g(y) over [0,3π4]\left[0, \frac{3\pi}{4}\right][0,43π​].2. Determine the value(s) of c c c in [0,3π4]\left[0, \frac{3\pi}{4}\right][0,43π​] that satisfy the Mean Value Theorem.

STEP 3

STEP 4

Evaluate the integral: ∫7sin⁡(y) dy=−7cos⁡(y)+C\int 7 \sin(y) \, dy = -7 \cos(y) + C∫7sin(y)dy=−7cos(y)+CThus, ∫03π47sin⁡(y) dy=[−7cos⁡(y)]03π4\int_0^{\frac{3\pi}{4}} 7 \sin(y) \, dy = \left[-7 \cos(y)\right]_0^{\frac{3\pi}{4}}∫043π​​7sin(y)dy=[−7cos(y)]043π​​

STEP 5

STEP 6

Calculate the average value: g(c)=13π4(722+7)g(c) = \frac{1}{\frac{3\pi}{4}} \left(\frac{7\sqrt{2}}{2} + 7\right)g(c)=43π​1​(272​​+7)Simplify: g(c)=43π(722+7)g(c) = \frac{4}{3\pi} \left(\frac{7\sqrt{2}}{2} + 7\right)g(c)=3π4​(272​​+7)

STEP 7

Determine the value(s) of c c c in [0,3π4]\left[0, \frac{3\pi}{4}\right][0,43π​] such that: g(c)=7sin⁡(c)=43π(722+7)g(c) = 7 \sin(c) = \frac{4}{3\pi} \left(\frac{7\sqrt{2}}{2} + 7\right)g(c)=7sin(c)=3π4​(272​​+7)

STEP 8

Solve for c c c: sin⁡(c)=43π(22+1)\sin(c) = \frac{4}{3\pi} \left(\frac{\sqrt{2}}{2} + 1\right)sin(c)=3π4​(22​​+1)Calculate the numerical value: sin⁡(c)≈43π(1.4142+1)\sin(c) \approx \frac{4}{3\pi} \left(\frac{1.414}{2} + 1\right)sin(c)≈3π4​(21.414​+1)

STEP 9

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Calculate the average value of $g(y)$ over $\left[0, \frac{3\pi}{4}\right]$ .
2. Determine the value(s) of $c$ in $\left[0, \frac{3\pi}{4}\right]$ that satisfy the Mean Value Theorem.

Evaluate the integral: $\int 7 \sin(y) \, dy = -7 \cos(y) + C$
Thus, $\int_0^{\frac{3\pi}{4}} 7 \sin(y) \, dy = \left[-7 \cos(y)\right]_0^{\frac{3\pi}{4}}$

Calculate the average value: $g(c) = \frac{1}{\frac{3\pi}{4}} \left(\frac{7\sqrt{2}}{2} + 7\right)$
Simplify: $g(c) = \frac{4}{3\pi} \left(\frac{7\sqrt{2}}{2} + 7\right)$

Determine the value(s) of $c$ in $\left[0, \frac{3\pi}{4}\right]$ such that: $g(c) = 7 \sin(c) = \frac{4}{3\pi} \left(\frac{7\sqrt{2}}{2} + 7\right)$

Solve for $c$ : $\sin(c) = \frac{4}{3\pi} \left(\frac{\sqrt{2}}{2} + 1\right)$
Calculate the numerical value: $\sin(c) \approx \frac{4}{3\pi} \left(\frac{1.414}{2} + 1\right)$