Find the average value of the function on the interval . Round your answer to 3 decimal places.
A)0.45
B)9.342
D)0.432
E)18

Question

Find the average value of the function  on the interval  . Round your answer to 3 decimal places.
A)0.45 
B)9.342 
D)0.432 
E)18

Quizplus · Accepted Answer

The average value of a function f(x) on an interval [a,b] is given by:
$$\frac{1}{b-a}\int_a^b f(x)\,dx$$

Using this formula, we need to find the definite integral of the given function on the interval [11eaa3e5_556f_3119_9f8f_6fd30c2cff65_TB5971_11,11eaa3e5_556f_0a08_9f8f_f35ea57751a0_TB5971_11]. The indefinite integral of the given function is:

$$\int11eaa3e5_556f_0a08_9f8f_f35ea57751a0_TB5971_11^{x^2-x+1}{dx}=\frac{1}{2}\int11eaa3e5_556f_0a08_9f8f_f35ea57751a0_TB5971_11^{2x-1}{d(2x-1)}=\frac{1}{2}\cdot\frac{11eaa3e5_556f_0a08_9f8f_f35ea57751a0_TB5971_11^{2x-1}}{2x-1}+C$$

So the definite integral on the given interval is:

$$\frac{1}{2}\cdot\left(\frac{11eaa3e5_556f_0a08_9f8f_f35ea57751a0_TB5971_11^{2(11eaa3e5_556f_0a08_9f8f_f35ea57751a0_TB5971_11)-1}}{2(11eaa3e5_556f_0a08_9f8f_f35ea57751a0_TB5971_11)-1}-\frac{11eaa3e5_556f_0a08_9f8f_f35ea57751a0_TB5971_11^{2(11eaa3e5_556f_3119_9f8f_6fd30c2cff65_TB5971_11)-1}}{2(11eaa3e5_556f_3119_9f8f_6fd30c2cff65_TB5971_11)-1}\right)$$

Simplifying the expression, we get:

$$\frac{1}{2}\cdot\left(\frac{11eaa3e5_556f_0a08_9f8f_f35ea57751a0_TB5971_11^{2\cdot11eaa3e5_556f_0a08_9f8f_f35ea57751a0_TB5971_11-1}}{2\cdot11eaa3e5_556f_0a08_9f8f_f35ea57751a0_TB5971_11-1}-\frac{11eaa3e5_556f_0a08_9f8f_f35ea57751a0_TB5971_11^{2\cdot11eaa3e5_556f_3119_9f8f_6fd30c2cff65_TB5971_11-1}}{2\cdot11eaa3e5_556f_3119_9f8f_6fd30c2cff65_TB5971_11-1}\right)$$

Now we just need to evaluate this expression and divide by the length of the interval (which is equal to the difference between the endpoints):

$$\frac{1}{11eaa3e5_556f_0a08_9f8f_f35ea57751a0_TB5971_11-11eaa3e5_556f_3119_9f8f_6fd30c2cff65_TB5971_11}\cdot\left(\frac{11eaa3e5_556f_0a08_9f8f_f35ea57751a0_TB5971_11^{2\cdot11eaa3e5_556f_0a08_9f8f_f35ea57751a0_TB5971_11-1}}{2\cdot11eaa3e5_556f_0a08_9f8f_f35ea57751a0_TB5971_11-1}-\frac{11eaa3e5_556f_0a08_9f8f_f35ea57751a0_TB5971_11^{2\cdot11eaa3e5_556f_3119_9f8f_6fd30c2cff65_TB5971_11-1}}{2\cdot11eaa3e5_556f_3119_9f8f_6fd30c2cff65_TB5971_11-1}\right)$$

Using a calculator, we get the value of the expression as approximately 11.666. Rounding this to 3 decimal places, we get the final answer as 11eaa3e5_556f_311a_9f8f_150cf2754b01_TB5971_11. Therefore, the correct choice is C.

Find the Average Value of the Function on the Interval