qu.1.topic=1-computing sums@ qu.1.1.question=Compute $\sum _{i=1}^{\mathrm{n}}\left(\mathrm{a}i+\mathrm{b}\right)$.@ qu.1.1.answer.num=\$answer@ qu.1.1.answer.units=@ qu.1.1.showUnits=false@ qu.1.1.grading=exact_value@ qu.1.1.negStyle=minus@ qu.1.1.numStyle=thousands @ qu.1.1.mode=Numeric@ qu.1.1.comment=The answer is \$answer.@ qu.1.1.editing=useHTML@ qu.1.1.algorithm=\$n=range(30,60,10); \$a=range(2,5); \$b=range(2,9); \$answer=\$b*\$n+\$a*\$n*(\$n+1)/2;@ qu.1.2.mode=Inline@ qu.1.2.editing=useHTML@ qu.1.2.algorithm=\$b=range(2,9); \$dx=switch(rint(4),0.1,0.2,0.4,0.5); \$xa=\$dx; \$xb=2*\$dx; \$xc=3*\$dx; \$xd=4*\$dx; \$fa=\$dx^2+\$b*\$dx; \$fb=4*\$dx^2+2*\$b*\$dx; \$fc=9*\$dx^2+3*\$b*\$dx; \$fd=16*\$dx^2+4*\$b*\$dx; \$answer=(30*\$dx^2+10*\$b*\$dx)*\$dx;@ qu.1.2.weighting=1,1,1@ qu.1.2.numbering=alpha@ qu.1.2.part.1.editing=useHTML@ qu.1.2.part.1.question=(Unset)@ qu.1.2.part.1.answer=\$xa, \$xb, \$xc, \$xd@ qu.1.2.part.1.mode=Ntuple@ qu.1.2.part.2.editing=useHTML@ qu.1.2.part.2.question=(Unset)@ qu.1.2.part.2.answer=\$fa, \$fb, \$fc, \$fd@ qu.1.2.part.2.mode=Ntuple@ qu.1.2.part.3.answer.units=@ qu.1.2.part.3.numStyle= @ qu.1.2.part.3.editing=useHTML@ qu.1.2.part.3.showUnits=false@ qu.1.2.part.3.question=(Unset)@ qu.1.2.part.3.mode=Numeric@ qu.1.2.part.3.grading=exact_value@ qu.1.2.part.3.negStyle=minus@ qu.1.2.part.3.answer.num=\$answer@ qu.1.2.question=Let $f\left(x\right)={x}^{2}+\mathrm{b}x$, set $\Delta x=\mathrm{dx}$, and let ${x}_{i}=i\Delta x$.

Enter ${x}_{1}$, ${x}_{2}$, ${x}_{3}$, ${x}_{4}$, separated by commas.
<1>

Enter $f\left({x}_{1}\right)$, $f\left({x}_{2}\right)$, $f\left({x}_{3}\right)$, $f\left({x}_{4}\right)$, separated by commas.
<2>

Compute $\sum _{i=1}^{4}f\left({x}_{i}\right)\text{\hspace{0.17em}}\Delta x$.
<3>  @ qu.2.topic=2-Riemann sums@ qu.2.1.mode=Inline@ qu.2.1.editing=useHTML@ qu.2.1.algorithm=\$a=range(1,3); \$n=range(2,3); \$c=range(2,9); \$dx=switch(rint(2),0.2,0.4); \$b=\$a+4*\$dx; \$xa=\$a+\$dx; \$xb=\$a+2*\$dx; \$xc=\$a+3*\$dx; \$xd=\$a+4*\$dx; \$fa=10*\$xa^\$n+\$c; \$fb=10*\$xb^\$n+\$c; \$fc=10*\$xc^\$n+\$c; \$fd=10*\$xd^\$n+\$c; \$answer=(\$fa+\$fb+\$fc+\$fd)*\$dx;@ qu.2.1.weighting=1,1,1@ qu.2.1.numbering=alpha@ qu.2.1.part.1.editing=useHTML@ qu.2.1.part.1.question=(Unset)@ qu.2.1.part.1.answer=\$xa, \$xb, \$xc, \$xd@ qu.2.1.part.1.mode=Ntuple@ qu.2.1.part.2.editing=useHTML@ qu.2.1.part.2.question=(Unset)@ qu.2.1.part.2.answer=\$fa, \$fb, \$fc, \$fd@ qu.2.1.part.2.mode=Ntuple@ qu.2.1.part.3.answer.units=@ qu.2.1.part.3.numStyle= @ qu.2.1.part.3.editing=useHTML@ qu.2.1.part.3.showUnits=false@ qu.2.1.part.3.question=(Unset)@ qu.2.1.part.3.mode=Numeric@ qu.2.1.part.3.grading=exact_value@ qu.2.1.part.3.negStyle=minus@ qu.2.1.part.3.answer.num=\$answer@ qu.2.1.question=Let   $f\left(x\right)=10{x}^{\mathrm{n}}+\mathrm{c}$.
List the evaluation points ${x}_{i}$ needed in a right-hand Riemann sum on the interval $\left[\mathrm{a},\mathrm{b}\right]$,
using $4$ points. Separate your answers with commas.

<1>

List the values $f\left({x}_{i}\right)$, separated by commas. Use the exact values.
<2>

Compute the Riemann sum. Use the exact value.
<3>  @ qu.2.2.mode=Inline@ qu.2.2.editing=useHTML@ qu.2.2.algorithm=\$a=range(1,3); \$n=range(2,3); \$c=range(2,9); \$dx=switch(rint(2),0.2,0.4); \$b=\$a+4*\$dx; \$xa=\$a; \$xb=\$a+\$dx; \$xc=\$a+2*\$dx; \$xd=\$a+3*\$dx; \$fa=10*\$xa^\$n+\$c; \$fb=10*\$xb^\$n+\$c; \$fc=10*\$xc^\$n+\$c; \$fd=10*\$xd^\$n+\$c; \$answer=(\$fa+\$fb+\$fc+\$fd)*\$dx;@ qu.2.2.weighting=1,1,1@ qu.2.2.numbering=alpha@ qu.2.2.part.1.editing=useHTML@ qu.2.2.part.1.question=(Unset)@ qu.2.2.part.1.answer=\$xa, \$xb, \$xc, \$xd@ qu.2.2.part.1.mode=Ntuple@ qu.2.2.part.2.editing=useHTML@ qu.2.2.part.2.question=(Unset)@ qu.2.2.part.2.answer=\$fa, \$fb, \$fc, \$fd@ qu.2.2.part.2.mode=Ntuple@ qu.2.2.part.3.answer.units=@ qu.2.2.part.3.numStyle= @ qu.2.2.part.3.editing=useHTML@ qu.2.2.part.3.showUnits=false@ qu.2.2.part.3.question=(Unset)@ qu.2.2.part.3.mode=Numeric@ qu.2.2.part.3.grading=exact_value@ qu.2.2.part.3.negStyle=minus@ qu.2.2.part.3.answer.num=\$answer@ qu.2.2.question=Let   $f\left(x\right)=10{x}^{\mathrm{n}}+\mathrm{c}$.
List the evaluation points ${x}_{i}$ needed in a left-hand Riemann sum on the interval $\left[\mathrm{a},\mathrm{b}\right]$,
using $4$ points. Separate your answers with commas.

<1>

List the values $f\left({x}_{i}\right)$, separated by commas. Use the exact values.
<2>

Compute the Riemann sum. Use the exact value.
<3>  @ qu.2.3.mode=Inline@ qu.2.3.editing=useHTML@ qu.2.3.algorithm=\$a=range(1,3); \$n=2; \$c=range(2,9); \$dx=switch(rint(2),0.2,0.4); \$b=\$a+4*\$dx; \$xa=\$a+.5*\$dx; \$xb=\$a+1.5*\$dx; \$xc=\$a+2.5*\$dx; \$xd=\$a+3.5*\$dx; \$fa=10*\$xa^\$n+\$c; \$fb=10*\$xb^\$n+\$c; \$fc=10*\$xc^\$n+\$c; \$fd=10*\$xd^\$n+\$c; \$answer=(\$fa+\$fb+\$fc+\$fd)*\$dx;@ qu.2.3.weighting=1,1,1@ qu.2.3.numbering=alpha@ qu.2.3.part.1.editing=useHTML@ qu.2.3.part.1.question=(Unset)@ qu.2.3.part.1.answer=\$xa, \$xb, \$xc, \$xd@ qu.2.3.part.1.mode=Ntuple@ qu.2.3.part.2.editing=useHTML@ qu.2.3.part.2.question=(Unset)@ qu.2.3.part.2.answer=\$fa, \$fb, \$fc, \$fd@ qu.2.3.part.2.mode=Ntuple@ qu.2.3.part.3.answer.units=@ qu.2.3.part.3.numStyle= @ qu.2.3.part.3.editing=useHTML@ qu.2.3.part.3.showUnits=false@ qu.2.3.part.3.question=(Unset)@ qu.2.3.part.3.mode=Numeric@ qu.2.3.part.3.grading=exact_value@ qu.2.3.part.3.negStyle=minus@ qu.2.3.part.3.answer.num=\$answer@ qu.2.3.question=Let   $f\left(x\right)=10{x}^{\mathrm{n}}+\mathrm{c}$.
List the evaluation points ${x}_{i}$ needed in a midpoint Riemann sum on the interval $\left[\mathrm{a},\mathrm{b}\right]$,
using $4$ points. Separate your answers with commas.

<1>

List the values $f\left({x}_{i}\right)$, separated by commas. Use the exact values.
<2>

Compute the Riemann sum. Use the exact value.
<3>  @ qu.3.topic=3-exact computation of quadratic integrals@ qu.3.1.mode=Inline@ qu.3.1.editing=useHTML@ qu.3.1.algorithm=\$a=range(1,10); \$c=2*\$a; \$b=range(2,5); \$F=mathml(3*x^2+2*\$a*x); \$dx=\$b/n; \$xi=\$b*i/n; \$fi=(3*\$b^2/n^2)*i^2+(2*\$a*\$b/n)*i; \$fidx=(3*\$b^3/n^3)*i^2+(2*\$a*\$b^2/n^2)*i; \$rsum=\$b^3*(n+1)*(2*n+1)/(2*n^2)+\$a*\$b^2*(n+1)/n; \$integral=\$b^3+\$a*\$b^2;@ qu.3.1.weighting=1,1,1,1,1,1@ qu.3.1.numbering=alpha@ qu.3.1.part.1.editing=useHTML@ qu.3.1.part.1.question=(Unset)@ qu.3.1.part.1.answer=\$dx@ qu.3.1.part.1.mode=Formula@ qu.3.1.part.2.editing=useHTML@ qu.3.1.part.2.question=(Unset)@ qu.3.1.part.2.answer=\$xi@ qu.3.1.part.2.mode=Formula@ qu.3.1.part.3.editing=useHTML@ qu.3.1.part.3.question=(Unset)@ qu.3.1.part.3.answer=\$fi@ qu.3.1.part.3.mode=Formula@ qu.3.1.part.4.editing=useHTML@ qu.3.1.part.4.question=(Unset)@ qu.3.1.part.4.answer=\$fidx@ qu.3.1.part.4.mode=Formula@ qu.3.1.part.5.editing=useHTML@ qu.3.1.part.5.question=(Unset)@ qu.3.1.part.5.answer=\$rsum@ qu.3.1.part.5.mode=Formula@ qu.3.1.part.6.answer.units=@ qu.3.1.part.6.numStyle=thousands @ qu.3.1.part.6.editing=useHTML@ qu.3.1.part.6.showUnits=false@ qu.3.1.part.6.question=(Unset)@ qu.3.1.part.6.mode=Numeric@ qu.3.1.part.6.grading=exact_value@ qu.3.1.part.6.negStyle=minus@ qu.3.1.part.6.answer.num=\$integral@ qu.3.1.question=Consider the right-hand Riemann sum representing the area under the graph of
\$F from $0$ to $b$ using $n$ equal subdivisions.

What is $\mathrm{\Delta x}$? (Your answer should be a function of $n$.)
<1>

What is the general formula for ${x}_{i}$ for the Riemann sum in terms of $n$?
<2>

What is the general formula for $f\left({x}_{i}\right)$ in terms of $n$?
<3>

What is the general formula for $f\left({x}_{i}\right)\text{\hspace{0.17em}}\mathrm{\Delta x}$ in terms of $n$?
<4>

Use the formulas $\sum _{i=1}^{n}i=\frac{n\left(n+1\right)}{2}$ and $\sum _{i=1}^{n}{i}^{2}=\frac{n\left(n+1\right)\left(2n+1\right)}{6}$
to compute the Riemann sum in terms of $n$.
<5>

Compute ${\int }_{0}^{\mathrm{b}}\left(3{x}^{2}+\mathrm{c}x\right)dx$ by taking a limit of the Riemann sum as $n\to \infty$.

<6>  @ qu.4.topic=4-estimating definite integrals from graphs@ qu.4.1.question=Determine ${\int }_{0}^{\mathrm{r}}f\left(x\right)dx$ using the graph of $f$ given below.
Note that you must use "abs" for the absolute value function.@ qu.5.2.answer=\$answer@ qu.5.3.mode=Formula Mod C@ qu.5.3.comment=The answer is \${mathml("\$answer")}.@ qu.5.3.editing=useHTML@ qu.5.3.algorithm=\$a=range(2,9); \$rone=rint(2); \$fone=switch(\$rone,sin(\$a*x),cos(\$a*x)); \$ansone=switch(\$rone,cos(\$a*x),sin(\$a*x)); \$sign=switch(\$rone,-1,1); \$ftwo=(sec x)^2; \$anstwo=tan x; \$answer=\$sign*(1/"\$a")*(\$ansone)+\$anstwo;@ qu.5.3.question=Find an antiderivative of   \${mathml(\$fone+\$ftwo)}.@ qu.5.3.answer=\$answer@ qu.5.4.mode=Formula Mod C@ qu.5.4.comment=The answer is \${mathml("\$answer")}. Note that the absolute value is optional because \${mathml(\$w)} $>0$.@ qu.5.4.editing=useHTML@ qu.5.4.algorithm=\$a=range(1,8); \$b=range(2,8); \$n=range(2,8,2); \$w=\$a+\$b*x^\$n; \$wpf=if(eq(\$n,2),x,x^(\$n-1)); \$wpc=\$n*\$b; \$function=(\$wpf)/(\$w); \$answer=("1"/(\$wpc))*ln(\$w);@ qu.5.4.question=Find an antiderivative of   \${mathml(\$function)}.@ qu.5.4.answer=\$answer@ qu.6.topic=6-definite integrals by the fundamental theorem@ qu.6.1.mode=Formula@ qu.6.1.comment=The answer is \${mathml("\$answer")}.@ qu.6.1.editing=useHTML@ qu.6.1.algorithm=\$a=range(1,2); \$b=range(2,4); \$c=range(2,9); condition:gt(\$b,\$a); \$r=rint(4); \$n=switch(\$r,1,2,-2,-3); \$m=switch(\$r,2,3,1,2); \$d=switch(\$r,2,3,\$a*\$b,2*\$a^2*\$b^2); \$function=if(eq(\$r,0),\$c*\$x,\$c*x^\$n); \$answer=(\$c*(\$b^\$m-\$a^\$m))/"\$d";@ qu.6.1.question=Compute ${\int \text{\hspace{0.17em}}}_{a}^{b}$ \${mathml(\$function)} \${mathml(d*x)}.@ qu.6.1.answer=\$answer@ qu.6.2.mode=Formula@ qu.6.2.comment=The answer is \${mathml("\$answer")}.@ qu.6.2.editing=useHTML@ qu.6.2.algorithm=\$b=range(2,9); \$c=range(2,9); \$a=range(-9,9); condition:not(eq(\$a*(\$a-1),0)); \$function=\$c*"e"^(\$a*x); \$answer="(\$c/\$a)*(e^(\$a*\$b)-1)";@ qu.6.2.question=Compute ${\int \text{\hspace{0.17em}}}_{0}^{b}$ \${mathml("\$function")} \${mathml(d*x)}.@ qu.6.2.answer=\$answer@ qu.6.3.mode=Formula@ qu.6.3.comment=The answer is \${mathml("\$answer")}.@ qu.6.3.editing=useHTML@ qu.6.3.algorithm=\$a=range(2,5); \$b=range(6,9); \$c=range(2,9); \$x=switch(rint(6),"t","u","v","w","y","z"); \$function=\$c/\$x; \$answer="\$c*ln(\$b/\$a)";@ qu.6.3.question=Compute ${\int \text{\hspace{0.17em}}}_{a}^{b}$ \${mathml(\$function)} \${mathml(d*\$x)}.@ qu.6.3.answer=\$answer@ qu.6.4.mode=Formula@ qu.6.4.comment=The answer is \${mathml("\$answer")}.@ qu.6.4.editing=useHTML@ qu.6.4.algorithm=\$a=range(2,5); \$b=range(1,5); \$n=range(2,5); \$r=range(0,2); \$t=range(2,4); condition:not(gt(\$t+\$n,6)); \$s=switch(\$r,1,1,-1); \$q=switch(\$r,7,1,\$t^\$n); condition:gt(\$s*\$b*\$q,\$s*\$a); \$w=switch(\$r,\$a+\$b*x^\$n,\$b*x^\$n-\$a,\$a-\$b*x^\$n); \$wpf=switch(\$r,if(eq(\$n,2),x,x^(\$n-1)),if(eq(\$n,2),x,x^(\$n-1)),if(eq(\$n,2),x,x^(\$n-1))); \$wpc=switch(\$r,\$n*\$b,\$n*\$b,-\$n*\$b); \$function=(\$wpf)/(\$w); \$from=switch(\$r,0,1,0); \$to=\$t; \$wto=switch(\$r,\$a+\$b*\$to^\$n,\$b*\$to^\$n-\$a,\$a-\$b*\$to^\$n); \$wfrom=switch(\$r,\$a,\$b-\$a,\$a); \$answer=("1"/(\$wpc))*"ln(\$wto/\$wfrom)";@ qu.6.4.question=Compute ${\int \text{\hspace{0.17em}}}_{from}^{to}$ \${mathml(\$function)} \${mathml(d*x)}.@ qu.6.4.answer=\$answer@ qu.7.topic=7-f values from f'@ qu.7.1.mode=Formula@ qu.7.1.comment=The answer is \${mathml("\$answer")}.@ qu.7.1.editing=useHTML@ qu.7.1.algorithm=\$c=range(1,9); \$n=range(2,5); \$np=\$n+1; \$a=range(1,4); \$b=range(-8,8); condition:not(gt(\$a+\$n,6)); \$function=\$c*x^\$n; \$C=(\$b*\$np-\$c*\$a^(\$np))/"\$np"; \$answer=(\$c/"\$np")*x^(\$np)+"\$C";@ qu.7.1.question=Find an antiderivative of   \${mathml("\$function")} that passes through the point $\left(a,b\right)$.@ qu.7.1.answer=\$answer@ qu.7.2.question=An object is originally located at $x=x$. Beginning at time zero, the object moves along the $x$ axis with velocity given by the graph below. Determine the location of the object at time $r$.
@ qu.7.2.answer.num=\$answer@ qu.7.2.answer.units=@ qu.7.2.showUnits=false@ qu.7.2.grading=exact_value@ qu.7.2.negStyle=minus@ qu.7.2.numStyle=thousands @ qu.7.2.mode=Numeric@ qu.7.2.comment=The net displacement is the integral of the velocity. The answer is \$answer.@ qu.7.2.editing=useHTML@ qu.7.2.algorithm=\$x=range(1,10); \$a=range(4,10); \$d=range(4,10); condition:eq(int((\$a+\$d)/2),(\$a+\$d)/2); \$c=range(1,2); \$s=range(2,3); \$q=range(1,4); \$m=\$q/(2*\$c); \$b=\$a+2*\$s*\$c; \$r=\$b+2*\$c+\$d; \$answer=\$x+\$q*(\$s*\$a+(\$s^2-1)*\$c-\$d); \$f=maple("seq([2*i,\$q*y,y=-1..\$s],i=1..18)");@ qu.7.3.mode=Formula@ qu.7.3.comment=The net displacement is the integral of the velocity. The answer is \${mathml("\$answer")}.@ qu.7.3.editing=useHTML@ qu.7.3.algorithm=\$x=range(-5,5); \$a=range(-5,5); \$b=range(-5,5); \$bm=-\$b; \$c=range(1,4); \$r=range(2,3); condition:not(eq(\$x*(\$a^2+\$b^2),0)); \$function=if(lt(\$b,0),\$a-\$bm*t+\$c*t^2,\$a+\$b*t+\$c*t^2); \$answer=(6*\$x+6*\$a*\$r+3*\$b*\$r^2+2*\$c*\$r^3)/"6";@ qu.7.3.question=An object is originally located at $x=x$. Beginning at time zero, the object moves along the $x$ axis with velocity given by \${mathml(\$function)}. Determine the exact location of the object at time $r$.@ qu.7.3.answer=\$answer@ qu.8.topic=8-average value@ qu.8.1.question=Determine the average value of $f$ on the interval $\left[0,r\right]$.
@ qu.8.1.answer.num=\$answer@ qu.8.1.answer.units=@ qu.8.1.showUnits=false@ qu.8.1.grading=toler_abs@ qu.8.1.err=.01@ qu.8.1.negStyle=minus@ qu.8.1.numStyle=thousands @ qu.8.1.mode=Numeric@ qu.8.1.comment=The answer is \$answer.@ qu.8.1.editing=useHTML@ qu.8.1.algorithm=\$a=range(4,10); \$d=range(4,10); condition:eq(int((\$a+\$d)/2),(\$a+\$d)/2); \$c=range(1,2); \$s=range(2,3); \$q=range(1,4); \$m=\$q/(2*\$c); \$b=\$a+2*\$s*\$c; \$r=\$b+2*\$c+\$d; \$answer=\$q*(\$s*\$a+(\$s^2-1)*\$c-\$d)/\$r; \$f=maple("seq([2*i,\$q*y,y=-1..\$s],i=1..18)");@ qu.8.2.mode=Formula@ qu.8.2.comment=The answer is \${mathml("\$answer")}.@ qu.8.2.editing=useHTML@ qu.8.2.algorithm=\$c=range(-4,4); condition:not(eq(\$c,0)); \$a=range(1,4); \$h=range(1,4); \$b=\$a+\$h; \$num=\$a^2+\$a*\$b+\$b^2; \$termtwo=\$c*(\$a+\$b); \$answer="\$num/3+\$termtwo";@ qu.8.2.question=Find the average value of   \${mathml(x^2+2*\$c*x)}   on the interval $\left[a,b\right]$.
Your answer should be exact.@ qu.8.2.answer=\$answer@ qu.9.topic=9-derivatives of integrals@ qu.9.1.mode=Formula@ qu.9.1.comment=The answer is \${mathml(\$answera)} \${mathml(\$answerb)}.@ qu.9.1.editing=useHTML@ qu.9.1.algorithm=\$a=range(0,2); \$b=range(2,9); \$c=range(2,9); \$d=range(2,9); \$m=range(2,6); \$n=range(2,6); \$f=switch(rint(3),"sin","cos","tan"); \$functiona=ln(\$b*t^\$n+\$c); \$functionb=\$f(\$d*t^-\$m); \$answera=ln(\$b*x^\$n+\$c); \$answerb=\$f(\$d*x^-\$m); \$answer=\$answera*\$answerb;@ qu.9.1.question=Find the derivative of ${\int }_{\mathrm{a}}^{x}$ \${mathml(\$functiona)} \${mathml(\$functionb)} $\text{\hspace{0.17em}}\mathrm{dt}$.@ qu.9.1.answer=\$answer@ qu.9.2.mode=Formula@ qu.9.2.comment=The answer is \${mathml(\$answer)}.@ qu.9.2.editing=useHTML@ qu.9.2.algorithm=\$a=range(0,2); \$b=range(2,9); \$n=range(2,6); \$v=switch(rint(5),"u","v","w","y","z"); \$function=sqrt(sec(\$b*\$v^\$n)); \$answer=sqrt(sec(\$b*x^\$n));@ qu.9.2.question=Find the derivative of ${\int }_{\mathrm{a}}^{x}$ \${mathml(\$function)} $\text{\hspace{0.17em}}\mathrm{dv}$.@ qu.9.2.answer=\$answer@