Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

What is the average value of 3x^(2)+4x on the interval 2 <= x <= 6 ?

What is the average value of 3x2+4x 3 x^{2}+4 x on the interval 2≤x≤6 2 \leq x \leq 6 ?

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Algebra 2right chevron icon
Average rate of changeright chevron icon

Full solution

Q. What is the average value of 3x2+4x 3 x^{2}+4 x on the interval 2≤x≤6 2 \leq x \leq 6 ?
  1. Average Value Formula: To find the average value of a function f(x)f(x) over the interval [a,b][a, b], we use the formula:\newlineAverage value = 1(b−a)∫abf(x) dx\frac{1}{(b-a)} \int_{a}^{b} f(x) \, dx\newlineHere, f(x)=3x2+4xf(x) = 3x^2 + 4x, a=2a = 2, and b=6b = 6.
  2. Definite Integral Calculation: First, we need to find the definite integral of f(x)f(x) from 22 to 66.∫26(3x2+4x) dx\int_{2}^{6} (3x^2 + 4x) \, dxThis requires us to integrate the function term by term.
  3. Term by Term Integration: The integral of 3x23x^2 with respect to xx is x3x^3, and the integral of 4x4x with respect to xx is 2x22x^2. So, ∫\int of (3x2+4x) dx=x3+2x2(3x^2 + 4x) \, dx = x^3 + 2x^2.
  4. Evaluation at Limits: Now we evaluate the antiderivative at the upper and lower limits of the interval and subtract: 63+2⋅626^3 + 2\cdot 6^2 - 23+2⋅222^3 + 2\cdot 2^2
  5. Upper Limit Calculation: Calculating the values:\newline(63+2×62)=(216+2×36)=(216+72)=288(6^3 + 2\times6^2) = (216 + 2\times36) = (216 + 72) = 288\newline(23+2×22)=(8+2×4)=(8+8)=16(2^3 + 2\times2^2) = (8 + 2\times4) = (8 + 8) = 16
  6. Subtract Lower Limit: Subtract the lower limit evaluation from the upper limit evaluation: 288−16=272288 - 16 = 272
  7. Average Value Calculation: Now, we divide this result by (b−a)(b - a) to find the average value:\newlineAverage value = 272(6−2)\frac{272}{(6 - 2)}
  8. Denominator Calculation: Calculating the denominator: 6−2=46 - 2 = 4
  9. Division Result: Dividing the result by the interval length:\newlineAverage value = 2724\frac{272}{4}
  10. Final Average Value: Simplifying the fraction:\newlineAverage value = 6868

More problems from Average rate of change

Question
We want to factor the following expression:\newlinex4+9 x^{4}+9 \newlineWhich pattern can we use to factor the expression?\newlineU U and V V are either constant integers or single-variable expressions.\newlineChoose 11 answer:\newline(A) (U+V)2 (U+V)^{2} or (U−V)2 (U-V)^{2} \newline(B) (U+V)(U−V) (U+V)(U-V) \newline(C) We can't use any of the patterns.
Get tutor helpright-arrow

Posted 3 months ago

Question
z=−19+3.14i z=-19+3.14 i \newlineWhat are the real and imaginary parts of z z ?\newlineChoose 11 answer:\newline(A)\newlineRe⁡(z)=−19 and Im⁡(z)=3.14 \begin{array}{l} \operatorname{Re}(z)=-19 \text { and } \\ \operatorname{Im}(z)=3.14 \end{array} \newline(B)\newlineRe⁡(z)=3.14i and Im⁡(z)=−19 \begin{array}{l} \operatorname{Re}(z)=3.14 i \text { and } \\ \operatorname{Im}(z)=-19 \end{array} \newline(c)\newlineRe⁡(z)=−19 and Im⁡(z)=3.14i \begin{array}{l} \operatorname{Re}(z)=-19 \text { and } \\ \operatorname{Im}(z)=3.14 i \end{array} \newline(D)\newlineRe⁡(z)=3.14 and Im⁡(z)=−19 \begin{array}{l} \operatorname{Re}(z)=3.14 \text { and } \\ \operatorname{Im}(z)=-19 \end{array}
Get tutor helpright-arrow

Posted 3 months ago

Question
z=−60−12iRe⁡(z)=Im⁡(z)= \begin{array}{l}z=-60-12 i \\ \operatorname{Re}(z)= \\ \operatorname{Im}(z)=\end{array}
Get tutor helpright-arrow

Posted 3 months ago

Question
z=−45i−15.5 z=-45 i-15.5 \newlineWhat are the real and imaginary parts of z z ?\newlineChoose 11 answer:\newline(A)\newlineRe⁡(z)=−45 and Im⁡(z)=−15.5 \begin{array}{l} \operatorname{Re}(z)=-45 \text { and } \\ \operatorname{Im}(z)=-15.5 \end{array} \newline(B)\newlineRe⁡(z)=−15.5 and Im⁡(z)=−45 \begin{array}{l} \operatorname{Re}(z)=-15.5 \text { and } \\ \operatorname{Im}(z)=-45 \end{array} \newline(C)\newlineRe⁡(z)=−45i and Im⁡(z)=−15.5 \begin{array}{l} \operatorname{Re}(z)=-45 i \text { and } \\ \operatorname{Im}(z)=-15.5 \end{array} \newline(D)\newlineRe⁡(z)=−15.5 and Im⁡(z)=−45i \begin{array}{l} \operatorname{Re}(z)=-15.5 \text { and } \\ \operatorname{Im}(z)=-45 i \end{array}
Get tutor helpright-arrow

Posted 3 months ago

Question
z=−12i+11 z=-12 i+11 \newlineWhat are the real and imaginary parts of z z ?\newlineChoose 11 answer:\newline(A)\newlineRe⁡(z)=11 and Im⁡(z)=−12 \begin{array}{l} \operatorname{Re}(z)=11 \text { and } \\ \operatorname{Im}(z)=-12 \end{array} \newline(B)\newlineRe⁡(z)=11 and Im⁡(z)=−12i \begin{array}{l} \operatorname{Re}(z)=11 \text { and } \\ \operatorname{Im}(z)=-12 i \end{array} \newline(C)\newlineRe⁡(z)=−12i and Im⁡(z)=11 \begin{array}{l} \operatorname{Re}(z)=-12 i \text { and } \\ \operatorname{Im}(z)=11 \end{array} \newline(D)\newlineRe⁡(z)=−12 and Im⁡(z)=11 \begin{array}{l} \operatorname{Re}(z)=-12 \text { and } \\ \operatorname{Im}(z)=11 \end{array}
Get tutor helpright-arrow

Posted 3 months ago

Question
(18+4i)+(−11+23i)= (18+4 i)+(-11+23 i)= \newlineExpress your answer in the form (a+bi) (a+b i) .
Get tutor helpright-arrow

Posted 3 months ago

Question
(14+60i)+(−30+2i)= (14+60 i)+(-30+2 i)= \newlineExpress your answer in the form (a+bi) (a+b i) .
Get tutor helpright-arrow

Posted 3 months ago

Question
(3−91i)+(67)= (3-91 i)+(67)= \newlineExpress your answer in the form (a+bi) (a+b i) .
Get tutor helpright-arrow

Posted 3 months ago

Question
(−71+2i)+(88−12i)= (-71+2 i)+(88-12 i)= \newlineExpress your answer in the form (a+bi) (a+b i) .
Get tutor helpright-arrow

Posted 3 months ago

Question
(−33−2i)−(50+9i)= (-33-2 i)-(50+9 i)= \newlineExpress your answer in the form (a+bi) (a+b i) .
Get tutor helpright-arrow

Posted 3 months ago

View all (3)