Merge pull request #943 from paulvojta/main

Bug fixes

Author: drdrew42

OpenProblemLibrary/Indiana/Indiana_setIntegrals3Definite/s4_4_27.pg

@@ -70,8 +70,6 @@ done term-by-term by rewriting the integral as:
   \int_{$d1}^{$c1} \frac{{$a1}x^2 + $b1}{\sqrt{x}} dx = 
     \int_{$d1}^{$c1} \left( \frac{{$a1}x^2}{\sqrt{x}} + \frac{$b1}{\sqrt{x}}  \right) dx
 \]
-  
-\]
 Then, we can find the derivative, which is:
 \[
   F(x) =  \frac{2}{5}\left(\frac{{$a1}x^3}{\sqrt{x}}\right)

OpenProblemLibrary/Indiana/Indiana_setSeries5IntegralTest/ur_sr_5_11.pg

@@ -144,7 +144,7 @@ $m = random(2,6,1);
 
 qa(~~@questions, ~~@answers,
 "\( \displaystyle \sum_{n=1}^\infty n e^{-$a n}  \)" , "CONV",
-"\( \displaystyle \sum_{n=1}^\infty n e^{$a n}  \)" , "DIV",
+"\( \displaystyle \sum_{n=1}^\infty n e^{$a n}  \)" , "NA",
 "\( \displaystyle \sum_{n=1}^\infty \frac{\ln{($d n)}}{n} \)" , "DIV",
 "\( \displaystyle \sum_{n=1}^\infty \frac{$b}{n \ln ($c n)} \)" , "DIV",
 "\( \displaystyle \sum_{n=1}^\infty \frac{$b}{n (\ln ($c n))^{$m}} \)" , "CONV",
@@ -213,18 +213,10 @@ EOT
 
 if ($slice[$i] == 1) {
 &SOLUTION(EV3(<<'EOT'));
-($j).  The function \(f(x) = x e^{$a x}\) is continuous and increasing because it is the product of 
-continuous increasing functions, so \(f(x) \geq f(1)=e^{$a}\) when \(1\leq x < \infty\).  Thus
-\[ \int_1^{\infty} f(x) \; dx \geq \int_1^{\infty} e^{$a} \; dx = \infty.\]  
-Since \(f\) is increasing it follows that 
-\[ f(n) \geq \int_{n-1}^n f(x)\; dx  \text{ for each } n=2,3,4,\cdots\]
-thus one can apply the integral test:
-\[ \begin{aligned} \sum_{n=1}^{\infty} f(n) &= f(1) + \sum_{n=2}^{\infty}f(n) \\ 
-   &\geq  f(1)+\sum_{n=2}^{\infty} \int_{n-1}^n f(x)\; dx  \\
-   &= f(1) + \int_1^{\infty} f(x)\; dx \\
-   & = \infty,
- \end{aligned}\]
-which shows that the series diverges.  
+($j).  The function \(f(x) = x e^{$a x}\) is increasing because it is
+the product of positive increasing functions.  Therefore it is not
+a decreasing function, so the Integral Test does not apply, and
+the correct answer is NA.
 
 EOT
 }

OpenProblemLibrary/Michigan/Chap4Sec7/Q29.pg

@@ -50,7 +50,7 @@ TEXT(beginproblem());
 Context()->texStrings;
 BEGIN_TEXT
 
-If \( x \), and \( y \) are both positive, evaluate
+If \( x \) and \( y \) are both positive, evaluate
 $PAR
 \(\displaystyle \lim_{p\rightarrow 0}\frac{\ln($a\, x^p + $b\, y^p)}{p} =\)
 \{ ans_rule(35) \}

OpenProblemLibrary/Rochester/setDerivatives2_5Implicit/s2_6_19_mo.pg

@@ -104,7 +104,7 @@ Use implicit differentiation to find an equation
 of the tangent line to the curve \( $F = $k \)
 at the point \( $P1 \).
 $BR $BR
-The \{ helpLink(equation) \} \{ans_rule(10) \}
+The \{ helpLink(equation) \} \{ans_rule(20) \}
 defines the tangent line to the curve at the
 point \( $P1 \).
 END_TEXT

OpenProblemLibrary/UVA-Stew5e/setUVA-Stew5e-C02S09-DerivAsFunct/2-9-24.pg

@@ -50,12 +50,8 @@ shortcuts!) to find the derivative of the function
 \[
     f(x) = \frac{$a x + $b}{$c x + $d}.
 \]
-Then state the domain of the function and the domain of the derivative.
-$BR
-$BBOLD Note: $EBOLD When entering interval notation in WeBWorK, use
-$BBOLD I $EBOLD for \(\infty\), $BBOLD -I $EBOLD for \(-\infty\),
-and $BBOLD U $EBOLD for the union symbol.  If the set is empty,
-enter "{}" without the quotation marks.
+Then, using \{ helpLink('interval notation') \}, state the domain of
+the function and the domain of the derivative.
 $BR
 $BR
 \(f'(x)\) = \{ans_rule(25) \}

OpenProblemLibrary/Utah/AP_Calculus_I/set9_Basic_Methods_of_Integration/1220s11p1.pg

@@ -28,7 +28,7 @@ TEXT(beginproblem());
 
 TEXT(EV2(<<EOT));
 $PAR
-\(\int \sin^2(3x) \hbox{d}x = \)   \{ans_rule(30)\}. 
+\(\int \sin^2(3x) \hbox{d}x = \)  \{ans_rule(30)\} \( {}+C \).
 EOT
 $ans = "-1/6 cos(3 x) sin(3 x) + x/2";
 ANS(fun_cmp($ans, limits=>[1,3], mode=>"antider", vars=>"x"));

OpenProblemLibrary/Utah/Calculus_II/set6_Indeterminate_Forms_and_Improper_Integrals/set6_pr15.pg

@@ -25,10 +25,11 @@ loadMacros(
 
 TEXT(beginproblem());
 $showPartialCorrectAnswers = 1;
-$e = 2.718;
+$e = E();
 
 TEXT(EV2(<<EOT));
-In all these problems, write I if the limit is either \(+\infty \) or \(-\infty \).
+In both of these problems, write I if the limit is either \(+\infty \)
+or \(-\infty \).
 $PAR
 Find the following limits:
 $PAR
@@ -37,7 +38,7 @@ $BR
 \{ans_rule(40)\}.
 $PAR
 
-(b) \( \lim_{x\rightarrow 0} x^{\sin(x)} =  \)
+(b) \( \lim_{x\rightarrow 0^{+}} x^{\sin(x)} =  \)
 $BR \{ans_rule(40)\}.
 $PAR
 

OpenProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/9_Introduction_to_Differential_Equations/9.5_First-Order_Linear_Equations/9.5.35.pg

@@ -48,7 +48,7 @@ Consider a series circuit consisting of a resistor of \(R\) ohms, an inductor of
 
 Find the solution to this equation with the initial condition \( I(0) = 0 \), assuming that \( R = $R \, \Omega \), \( L = $L \) H, and \( V(t) \) is constant with \( V(t) = $V \) V.
 $PAR
-\( V(t) = \) \{ans_rule()\}
+\( I(t) = \) \{ans_rule()\}
 END_TEXT
 Context()->normalStrings;
 

OpenProblemLibrary/ma122DB/set10/s4_10_75.pg

@@ -44,7 +44,7 @@ Answer: \{ans_rule(30) \}
 END_TEXT
 
 $t1 = -$v1/32;
-$ans =   "$v1*$v1/64";
+$ans =   $v1*$v1/64;
 ANS(num_cmp($ans));
 
 
@@ -55,10 +55,10 @@ position at time \( t \) (\( s(t) \)), instantaneous velocity
 at time \( t \) (\( v(t) \)), and acceleration at time \( t\) (\(a(t)\)).
 This relationship is given by:
 $BR$BR
-\[  \begin{array}
-    v(t)  = s'(t) \\
-    a(t)  = v'(t) = s''(t) \\
-  \end{array}
+\[  \begin{split}
+    v(t) &= s'(t) \\
+    a(t) &= v'(t) = s''(t) \\
+  \end{split}
 \]
 $BR$BR
 With this in mind, this problem becomes an exercise in finding
@@ -92,11 +92,11 @@ and we know the stone struck the ground (so its position was 0) at
 \( t = !{$t1:%5.3f} \), we can solve the following equation for \( C_2 \).
 $BR$BR
 \[
-  \begin{array}
-    s(!{$t1:%5.3f}) =  0 \\
-    -16(!{$t1:%5.3f})^2 + C_2 = 0 \\
-    C_2 = !{$ans:%5.3f} \\
-  \end{array}
+  \begin{split}
+    s(!{$t1:%5.3f}) &= 0 \\
+    -16(!{$t1:%5.3f})^2 + C_2 &= 0 \\
+    C_2 &= !{$ans:%5.3f} \\
+  \end{split}
 \]
 Now, since we want to know the height from which the stone was dropped, we 
 just need to find its position at time \( t = 0 \).  But this is \( s(0) = -16(0)^2+ $ans \)

OpenProblemLibrary/ma122DB/set8/s4_3_27.pg

@@ -38,7 +38,8 @@ $showPartialCorrectAnswers = 0;
 BEGIN_TEXT
 $BR$BR \{image("s4_3_27.gif")\} $BR$BR
 For the function \(f\) given above, determine whether the following conditions
-are true. Input $BITALIC T $EITALIC if the condition is ture, otherwise input $BITALIC F $EITALIC . $BR$BR
+are true. Input $BITALIC T $EITALIC if the condition is true,
+otherwise input $BITALIC F $EITALIC . $BR$BR
 (a) \( f'(x)<0 \) if \(0<x<2\);  \{ans_rule(10) \} $BR
 (b) \( f'(x)>0 \) if \(x>2\);  \{ans_rule(10) \} $BR
 (c) \( f''(x)<0 \) if \(0\le x<1\);  \{ans_rule(10) \} $BR