uni

commit 5c276d6f15b18442fec68c98be1998513e489d19
parent e8cf90bd263022333d81e666d3b190f95a525d21
Author: thing1 <thing1@seacrossedlovers.xyz>
Date:   Tue, 18 Nov 2025 22:50:57 +0000

homework

Diffstat:
A
MP10610/hw3/Makefile
 | 
2
++
A
MP10610/hw3/hw3.ms
 | 
151
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

2 files changed, 153 insertions(+), 0 deletions(-)
diff --git a/MP10610/hw3/Makefile b/MP10610/hw3/Makefile
@@ -0,0 +1,2 @@
+hw3.pdf: hw3.ms
+	groff -e -Tps -ms hw3.ms | ps2pdf - > hw3.pdf
diff --git a/MP10610/hw3/hw3.ms b/MP10610/hw3/hw3.ms
@@ -0,0 +1,151 @@
+.TL
+Calculus assignment 3
+.AU
+Lucas Standen
+.AI
+Aberystwyth University
+.2C
+
+.EQ
+delim @@
+.EN
+.EQ
+delim @#
+.EN
+
+.LP
+2) The number of bee's after 15 days
+
+4)
+
+@ int f''(x) dx #
+
+@ int x sup {-3 over 2} dx #
+
+@ 2 over sqrt x + C #
+
+@ C = 3 #
+
+@ int 2x sup {-1 over 2} + 3 dx #
+
+@ 4 sqrt x + 3x + C = 0 #
+
+@ C = 0 #
+
+@ y = 4 sqrt x + 3x #
+
+5)d)
+
+@ int from 0 to a x sqrt {a sup 2 - x sup 2} dx#
+
+let @ u = a sup 2 - x sup 2 #
+
+@ du over dx = - 2x #
+
+@ - 1 over {2x} du = dx #
+
+@ u = a sup 2 - a sup 2 # Adjust the bounds
+
+@ u = 0 #
+
+@ int from 0 to 0 sqrt u du #
+
+@ 0 # Any integral from 0 to 0 is 0
+
+5)f)
+
+@ int 1 over {a sup 2 + x sup 2} #
+
+let @ u = x over a #
+
+@ du over dx = 1 over a #
+
+@ 1 over a int 1 over {a sup 2 + 1} du #
+
+@ 1 over a ln(u sup 2 + 1) 2u #
+
+@ { ln({x over a + 1}) } over a {2x over { a sup 2 }} #
+
+6)b)
+
+@ int 8x ln(x) dx #
+
+@ u = ln(x) #
+
+@ u' = 1 over x #
+
+@ v' = 8x #
+
+@ v = 4x sup 2 #
+
+@ 4x sup 2 ln(x) - int 1 over x 4x sup 2 dx #
+
+@ 4x sup 2 ln(x) - int 1 4x dx #
+
+@ 4x sup 2 ln(x) - 2x sup 2 + C #
+
+6)e)
+
+@ int from 1 to 4 e sup {sqrt x} dx #
+
+let @ u = sqrt x #
+
+@ du over dx = 1 over 2 x sup { - 1 over 2 } #
+
+@ du over dx = 1 over { 2 sqrt x } #
+
+@ 2 int from {1 over 2} to {1 over 4} u e sup 2 du #
+
+@ u = u #
+
+@ u' = 1 #
+
+@ v' = e sup u #
+
+@ v = e sup u #
+
+@ 2 (u e sup u - int from {1 over 2} to {1 over 4} e sup u du) #
+
+@ 2 e sup {sqrt x} ( sqrt x - 1 ) #
+
+@ [ from {1 over 2} to {1 over 4} 2 e sup {sqrt x} ( sqrt x - 1 ) ] #
+
+@ - sqrt e - e sup { { sqrt 2} over 2 } ( - 2 + sqrt 2 ) #
+
+7)a)
+
+@ y = tan sup -1 x #
+
+@ x = tan y #
+
+@ tan(x) = tan(tan sup -1 x) #
+
+@ d over dx tan y = d over dx x #
+
+@ sec sup 2 y dy over dx = 1 #
+
+@ 1 over { sec sup 2 y} = dy over dx #
+
+@ 1 over { 1 + tan sup 2 y } = dy over dx #
+
+@ 1 over {1 + x sup 2} = dy over dx #
+
+@ int from 0 to 1 tan sup -1 x dx #
+
+@ u = tan sup -1 x #
+
+@ u' = 1 over {1 + x sup 2} #
+
+@ v' = 1 #
+
+@ v = x #
+
+@ x tan sup -1 x - int from 0 to 1 x over {1 + x sup 2 } dx #
+
+@ x tan sup -1 x - [ from 0 to 1 x tan sup -1 x ] #
+
+@ [ from 0 to 1 x tan sup -1 x - 1 over {4 pi} ] #
+
+@ 1 over {4 pi} - 1 over {4 pi} #
+
+@ 0 #