CALC 2414 EXAM 1 Questions And Complete
Solutions (Verified Solutions) 2026
R^3, - CORRECT ANSWER -- x=4 denotes an entire plane in R^3 parallel to yz plane
- x+y=2 is an infinite plane in R^3 where z is infinite
- z=y^2 is a parabola shaped plane along x
- x^2+z^2=9 is a circle shaped parabola along y axis
- x^2+y^2<4 now rather than plane its a solid(cylinder going along z axis) bc of the inequality its
a solid region not including the boundary of the circle
distance formula - CORRECT ANSWER -d=√(x2-x1)^2+(y2-y1)^2
distance of sphere can be found with: r^2= (x2-x1)^2+(y2-y1)^2 +(z2-z1)^2
finding the equation of a sphere w/ the origin(or center) and a point - CORRECT ANSWER -Use
the center as x1, y1, z1, and the given point as x2, y2, z2. then use the distance equation to find r,
then plug in the center into the equation of a sphere
- sometimes u may be given a long equation in which u need to complete the square to find
answer
vector things to rmbr - CORRECT ANSWER -- if finding vector PQ the answer is <Q-P> for
each x,y,z. its where its going minus where it came from
-component forms means <x,y,z>
vectors - CORRECT ANSWER -- vector addition, scalar multiplication
-given vector from A(x1,y1) to B(x2,y2) vector A(AB->) is <x2-x1, y2-y1> can also have z at
end of 3D
-length of vectors can be found with |a|= √ax^2+ay^2+az^2
, - when asked to add, subtract, multiply vector points: 1<x,y>, 2<a,b> |1+2|=? <x+a, y+b> then
use magnitude equation to find length
- standard basic vectors: i<1,0,0> , j<0,1,0> , k<0,0,1>
unit vectors: length of vector is 1. u= a/|a|
ex) a=2i-j-2k find unit vector. |a|= √2^2+(-1)^2+(-2)^2 = 3 so 2i-j-2k/3= 2/3i - 1/3j - 2/3k (w/
hat)
∫secxdx
∫cscxdx - CORRECT ANSWER -= ln |secx + tanx| + C
= -ln |cscx + cotx| + C
∫tanxdx
∫cotxdx - CORRECT ANSWER -= ln|secx| + C
=ln|sinx| + C
∫1/x^2 + a^2
∫1/√a^2-x^2 - CORRECT ANSWER -1/a arctan (x/a) + C
arcsin(x/a) + C
half angle identities - CORRECT ANSWER -Sin^2(x)=1/2(1-cos(2x))
Cos^2(x)=1/2(1+cos(2x))
Solutions (Verified Solutions) 2026
R^3, - CORRECT ANSWER -- x=4 denotes an entire plane in R^3 parallel to yz plane
- x+y=2 is an infinite plane in R^3 where z is infinite
- z=y^2 is a parabola shaped plane along x
- x^2+z^2=9 is a circle shaped parabola along y axis
- x^2+y^2<4 now rather than plane its a solid(cylinder going along z axis) bc of the inequality its
a solid region not including the boundary of the circle
distance formula - CORRECT ANSWER -d=√(x2-x1)^2+(y2-y1)^2
distance of sphere can be found with: r^2= (x2-x1)^2+(y2-y1)^2 +(z2-z1)^2
finding the equation of a sphere w/ the origin(or center) and a point - CORRECT ANSWER -Use
the center as x1, y1, z1, and the given point as x2, y2, z2. then use the distance equation to find r,
then plug in the center into the equation of a sphere
- sometimes u may be given a long equation in which u need to complete the square to find
answer
vector things to rmbr - CORRECT ANSWER -- if finding vector PQ the answer is <Q-P> for
each x,y,z. its where its going minus where it came from
-component forms means <x,y,z>
vectors - CORRECT ANSWER -- vector addition, scalar multiplication
-given vector from A(x1,y1) to B(x2,y2) vector A(AB->) is <x2-x1, y2-y1> can also have z at
end of 3D
-length of vectors can be found with |a|= √ax^2+ay^2+az^2
, - when asked to add, subtract, multiply vector points: 1<x,y>, 2<a,b> |1+2|=? <x+a, y+b> then
use magnitude equation to find length
- standard basic vectors: i<1,0,0> , j<0,1,0> , k<0,0,1>
unit vectors: length of vector is 1. u= a/|a|
ex) a=2i-j-2k find unit vector. |a|= √2^2+(-1)^2+(-2)^2 = 3 so 2i-j-2k/3= 2/3i - 1/3j - 2/3k (w/
hat)
∫secxdx
∫cscxdx - CORRECT ANSWER -= ln |secx + tanx| + C
= -ln |cscx + cotx| + C
∫tanxdx
∫cotxdx - CORRECT ANSWER -= ln|secx| + C
=ln|sinx| + C
∫1/x^2 + a^2
∫1/√a^2-x^2 - CORRECT ANSWER -1/a arctan (x/a) + C
arcsin(x/a) + C
half angle identities - CORRECT ANSWER -Sin^2(x)=1/2(1-cos(2x))
Cos^2(x)=1/2(1+cos(2x))
