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A collection of calculus problems for a final exam, covering topics such as finding equations of spheres, angles between vectors, projections, unit vectors, vectors orthogonal to given vectors, parametric equations of lines, intersections of lines and planes, acceleration, length of curves, curvature, and more. Students preparing for a calculus 3 exam are encouraged to review these problems for additional practice.
Typology: Exercises
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Calculus 3 Final Exam Review Packet
Ans: (x-1)
2
+(y-2)
2
+(z-3)
2
Ans: ฮธ=ฯ/
Projection of b onto a = (a * b/ |a|
2
)a
Ans: 18/25 <3,4,0>
U= - a/|a|
Ans: <1,-2,-1>/โ 6
Ans: <3,3,-3>
6a. Find parametric equations of a line and symmetric equations through point.
P<2,-1,7> with direction vector V<3,-2,6>
x=2+3t Symmetric
y=- 1 - 2t Parametric (x-2)/3 = (y+1)/- 2 =(z-7)/
z=7+6t
Ans: <0,1,2>
v(0)= i - j + k and initial position vector r(0)=2k
a. Find the velocity vector at time t.
b. Find the speed of the particle at time = 1
c. Find the position vector at time t.
t
,e
t
sin(t),e
t
cos(t)> 0<t<
Hint: Arc Length = โซ โ๐โฒ(๐ก)
2
2
2
๐
๐
dt
Ans: โ 3 (e-1)
2
+4,2t-3,0>
Hint: Curvature = |rโ(t) x rโ(t)| / |rโ(t)|
3
Ans: 4/2(t
2
3/
unit tangent vector.
Ans: x= โ 3 /2+1/2t, y=1/2-โ 3 /2t, z=
Tangent Vector: <cos(t),-sin(t),0> / 1
c. z-8=x
2
2
+4y
Ans: Elliptic paraboloid centered at (3,-2,-5)
๐ฅ,๐ฆโ> 0 , 0
๐ฅ
2
+๐ฆ
2
โ๐ฅ
2
+๐ฆ
2
Ans: 18
๐ฅ,๐ฆโ 0 , 0
๐ฅ
4
โ 4 ๐ฆ
4
๐ฅ
2
2
Ans: 0
๐ฅ,๐ฆโ 0 , 0
๐ฅ
2
๐ฅ
2
+๐ฆ
2
x
ln ๐ฆ at (3,1,0).
Ans: z=e
3
(y-1)
a. z=tan(3x
2
y
5
b. xz
4
+3yz=x
2
y
3
is decreasing at a rate of 2.5 in/sec. At what rate is the volume of the cone changing when
radius is 120 in. and height 140in.? Remember dv= dv/dr * dr/dt + dv/dh * dh/dt
Ans: 8160 ฯ in
3
/sec
2
y-xy
2
Remember D=f xx
*f yy
2
Ans: f(0,0) = 0 saddle point, f(3,0)=0 saddle point, f(0,3) saddle point, f(1,1)=1 local max
2
+2y
2
+3z
2
Ans: Max:
2 โ
๐ฅ
3
, Min: -
2 โ
๐ฅ
3
Ans:
243 ๐
5
Ans:
16
9
2
2
that lies above the xy plane.
Hint: Surface Area= โฌ
2
2
Ans:
๐
6
2
that lies above the triangle in
the xy plane with vertices (0,0), (0,1), (2,1).
Ans:
1
6
2
2
dV where E is the solid
bounded by paraboloids z=x
2
+y
2
and z=16-3x
2
2
in cylindrical coordinates.
Ans:
2
2
2
where E is bounded by cone ฮธ=
๐
6
and above ฯ= 2 in spherical
coordinates.
Ans:
2
2
dV where E is the solid
bounded by paraboloids z=x
2
+y
2
and z= 16 - 3x
2
2
in cylindrical coordinates.
Ans:
2
+y
2
and z=18-x
2
2
Ans:
(0,1,0), (1,1,0), (0,1,1). Hint: Set up the equation for a plane to solve for bounds.
Ans:
4
๐ฆ dS where C is the upper half of the circle x
2
+y
2
Ans:
128
5
Remember: โซ
๐(๐ฅ, ๐ฆ) ds = โซ
2
2
dt
๐ฅ๐ง ds x=6t, y= โ
2 t
2
, z=2t
3
and 0<t<
Ans: (
12
35
gradiet f = F.
Ans:
๐๐
๐๐ฆ
= cos(๐ฅ) + sin(๐ฆ),
๐๐
๐๐ฅ
= cos(๐ฅ) + sin (๐ฆ), and f(x,y)=ycos(x)-cos(y)+C
๐น โ ๐๐ F(x,y)= y i + x j and C is the curve y=x
4
3
from (1,0) to (2,8).
Ans: 16
