Finding lower and upper sums for
$f\left(x\right)=\text{sin}\phantom{\rule{0.1em}{0ex}}x$
Find a lower sum for
$f\left(x\right)=\text{sin}\phantom{\rule{0.1em}{0ex}}x$ over the interval
$\left[a,b\right]=\left[0,\frac{\pi}{2}\right];$ let
$n=6.$
Let’s first look at the graph in
[link] to get a better idea of the area of interest.
The intervals are
$\left[0,\frac{\pi}{12}\right],\left[\frac{\pi}{12},\frac{\pi}{6}\right],\left[\frac{\pi}{6},\frac{\pi}{4}\right],\left[\frac{\pi}{4},\frac{\pi}{3}\right],\left[\frac{\pi}{3},\frac{5\pi}{12}\right],$ and
$\left[\frac{5\pi}{12},\frac{\pi}{2}\right].$ Note that
$f\left(x\right)=\text{sin}\phantom{\rule{0.1em}{0ex}}x$ is increasing on the interval
$\left[0,\frac{\pi}{2}\right],$ so a left-endpoint approximation gives us the lower sum. A left-endpoint approximation is the Riemann sum
$\sum _{i=0}^{5}\text{sin}\phantom{\rule{0.1em}{0ex}}{x}_{i}\left(\frac{\pi}{12}\right)}.$ We have
Using the function
$f\left(x\right)=\text{sin}\phantom{\rule{0.1em}{0ex}}x$ over the interval
$\left[0,\frac{\pi}{2}\right],$ find an upper sum; let
$n=6.$
The use of sigma (summation) notation of the form
$\sum _{i=1}^{n}{a}_{i}$ is useful for expressing long sums of values in compact form.
For a continuous function defined over an interval
$\left[a,b\right],$ the process of dividing the interval into
n equal parts, extending a rectangle to the graph of the function, calculating the areas of the series of rectangles, and then summing the areas yields an approximation of the area of that region.
The width of each rectangle is
$\text{\Delta}x=\frac{b-a}{n}.$
Riemann sums are expressions of the form
$\sum _{i=1}^{n}f\left({x}_{i}^{*}\right)\text{\Delta}x},$ and can be used to estimate the area under the curve
$y=f\left(x\right).$ Left- and right-endpoint approximations are special kinds of Riemann sums where the values of
$\left\{{x}_{i}^{*}\right\}$ are chosen to be the left or right endpoints of the subintervals, respectively.
Riemann sums allow for much flexibility in choosing the set of points
$\left\{{x}_{i}^{*}\right\}$ at which the function is evaluated, often with an eye to obtaining a lower sum or an upper sum.
State whether the given sums are equal or unequal.
$\sum _{i=1}^{10}i$ and
$\sum _{k=1}^{10}k$
$\sum _{i=1}^{10}i$ and
$\sum _{i=6}^{15}\left(i-5\right)$
$\sum _{i=1}^{10}i\left(i-1\right)$ and
$\sum _{j=0}^{9}\left(j+1\right)j$
$\sum _{i=1}^{10}i\left(i-1\right)$ and
$\sum _{k=1}^{10}\left({k}^{2}-k\right)$
a. They are equal; both represent the sum of the first 10 whole numbers. b. They are equal; both represent the sum of the first 10 whole numbers. c. They are equal by substituting
$j=i-1.$ d. They are equal; the first sum factors the terms of the second.
tangent line at a point/range on a function f(x) making f'(x)
Luis
Principles of definite integration?
ROHIT
For tangent they'll usually give an x='s value. In that case, solve for y, keep the ordered pair. then find f(x) prime. plug the given x value into the prime and the solution is the slope of the tangent line. Plug the ordered pair into the derived function in y=mx+b format as x and y to solve for B
Anastasia
parcing an area trough a function f(x)
Efrain
Find the length of the arc y = x^2 over 3 when x = 0 and x = 2.
what's career can one specialize in by doing pure maths
Lucy
Lucy Omollo...... The World is Yours by specializing in pure math. Analytics, Financial engineering ,programming, education, combinatorial mathematics, Game Theory. your skill-set will be like water a necessary element of survival.
mathematics seems to be anthropocentric deductive reasoning and a little high order logic. I only say this because I can only find two things going on which is infinitely smaller than 0 and anything over 1
David
More comprehensive list here: ***onetonline.org/find/descriptor/result/1.A.1.c.1?a=1
Bruce
so how can we differentiate inductive reasoning and deductive reasoning
Henry
thanks very much Mr David
Henry
hi everyone
Sabir
is there anyone who can guide me in learning the mathematics easily
Sabir
Hi Sabir
first step of learning mathematics is by falling in love with it and secondly, watch videos on simple algebra then read and solved problems on it
Leo
yes sabir just do every time practice that is the solution
Henry
it will be work over to you ,u know how mind work ,it prossed the information easily when u are practising regularly