useful equations
1. A line that passes through the point P0 (x0 , y0 , z0 ) and is parallel to the vector v = ha, b, ci
has the vector parametrization
−−→
r(t) = OP0 + t v,
and the parametric equations
x = x0 + a t
y = y0 + b t .
z = z0 + c t
2. Projection of u along v: proj v u = u|| =
u · v
v·v
v.
3. Orthogonal projection of u: orth v u = u⊥ = u − proj v u = u − u|| .
4. The plane that passes through the point P0 (x0 , y0 , z0 ) with normal vector n = ha, b, ci has
the vector equation
n · hx, y, zi = d,
and the scalar equation
ax + by + cz = d
where d = n · hx0 , y0 , z0 i = ax0 + by0 + cz0 .
5. Fundamental Theorem of Calculus for vector-valued functions:
Z b
r(t)dt = R(b) − R(a)
a
where R(t) is the antiderivative of r(t).
Z t
Z t
|v(u)| du.
|r0 (u)| du =
6. Arc length function: s(t) =
a
a
7. Rate of change of arc length:
ds
= |r0 (t)| = |v|
dt
8. Unit tangent vector: T =
r0 (t)
v
=
0
|r (t)|
|v|
9. Unit normal vector: N =
T0 (t)
|T0 (t)|
10. Unit binormal vector: B = T × N
11. Curvature: κ = |T0 (s)| =
|T0 (t)|
|T0 (t)|
|r0 (t) × r00 (t)|
|v × a|
=
=
=
0
0
3
|r (t)|
|v|
|r (t)|
|v|3
12. Radius of the osculating circle or radius of curvature: ρ =
13. Acceleration: a = v0 (t) = r00 (t) = aT T + aN N
14. aT = |v(t)|0 =
v·a
|v|
15. aN = κ|v|2 =
|v × a|
=
|v|
q
|a|2 − a2T
1
κ
16. Tangent plane to the surface z = f (x, y) at the point (x0 , y0 , z0 ):
z − z0 = fx (x0 , y0 )(x − x0 ) + fy (x0 , y0 )(y − y0 )
17. Linearization of f (x, y, z) at (a, b, c):
L(x, y, z) = f (a, b, c) + fx (a, b, c)(x − a) + fy (a, b, c)(y − b) + fz (a, b, c)(z − c)
18. Linear approximation, or tangent plane approximation, of f (x, y, z) at (a, b, c):
f (x, y, z) ≈ f (a, b, c) + fx (a, b, c)(x − a) + fy (a, b, c)(y − b) + fz (a, b, c)(z − c)
19. Total differential of w = f (x, y, z):
dw = fx (x, y, z) dx + fy (x, y, z) dy + fz (x, y, z) dz =
∂z
∂z
∂z
dx +
dy +
dz
∂x
∂y
∂z
20. Chain rule: Suppose u is a differentiable function of the n variables x1 , x2 , . . . , xn and each xj
is a differentiable function of the m variables t1 , t2 , . . . , tm . Then u is a function of t1 , t2 , . . . , tm
and
∂u
∂u ∂x1
∂u ∂x2
∂u ∂xn
=
+
+ ··· +
∂ti
∂x1 ∂ti
∂x2 ∂ti
∂xn ∂ti
for each i = 1, 2, . . . , m.
√
1. What is the domain of the vector function r(t) = et , ln t, t ?
2. Find the center and radius of the circle with parametrization r(t) = (2+3 cos t) i+3 sin t j+5 k.
3. Does the point (4, 3, 0) lie on the curve r(t) = t2 , 1 − t, sin(πt) ? Show your work.
4. The position of a particle is r(t) = 4t − t2 , 3 − t . Calculate v(1) and a(1). Plot both vectors
on the graph so their basepoints (or tails) are at the terminal (or end) point of r(1). Label
the vectors.
5. Give a trigonometric parametrization of the circle of radius 4 in the xy-plane, centered at
the origin, oriented clockwise. The point (0, 4) should correspond to t = 0. Use t as the
parameter.
6. Give a parameterization of
p the curve that is the intersection between the parabolic cylinder
3
y = x and the cone z = x2 + y 2 . Use t as the parameter.
7. Find parametric equations of the line tangent to the curve r(t) = t2 − 1, t2 + 1, t + 1 at
t = 1. Show your work.
8. This graph shows the trace of f (x, y) = ax2 + by in the vertical plane y = −1. Find values
for a and b. Show your work. This is a high school algebra problem!
9. Suppose r(t) = t2 , 1 − t, 4t , a(2) = h1, 3, 3i, and a0 (2) = h−1, 4, 1i. Calculate the derivative
of r(t) · a(t) at t = 2. Show your work.
10. Many circles, with different radiuses and centers, can pass through the points (2, 0) and
(−2, 0). What is the largest curvature these circles can have? The smallest?
√
11. On a particular curve the rate of change of r(t) is r0 (t) = 2 cos 2t, 2 sin 2t, t . Find the
length of the curve between t = 0 and t = 1. Show your work.
a
. Treating a
+4
as a variable, use calculus to find the value of a that maximizes κ. Show your work. This is
a Calculus I problem!
12. The helix r(t) = ha cos t, a sin t, 2ti (where a ≥ 0) has the curvature κ =
a2
13. Find aT for the particle with the position vector r(t) = cos t, sin t, t2 . Show your work.
14. Find the limit or show it does not exist:
x2 − y 2
. Show your work.
(x,y)→(0,0) x2 + y 2
lim
15. Find aN at t = 1 for the particle with the velocity vector v(t) = 4 i + 3 j + t k. Show your
work.
16. Find the limit or show it does not exist:
sin(πy)
. Show your work.
xy
(x,y)→(2,3)
lim
17. Find the velocity vector for the particle that has a(t) = t i + et j + e−t k and the initial velocity
v(0) = j + k. Show your work.
√
√
x+ y
18. Plot the domain of f (x, y) = √
. Explain your reasoning. Label the axes and tic
4−x−y
marks and shade in the domain.
19. The curve r(t) lies in the plane x + 2y − 2z = 4. Show that r0 (t) is perpendicular to the
plane’s normal vector. Hint: write the equation of the plane in vector form. Show your work.
20. The partial derivative of f (x, y) = cos(x) sin(2y) with respect to x is (circle one)
(a) − sin(x) sin(2y) + 2 cos(x) cos(2y).
(b) sin(x) sin(2y) + 2 cos(x) cos(y).
(c) − sin(x) sin(2y).
(d) sin(x) sin(2y).
21. A particle travels around a circle. The unit tangent vector T and acceleration vector a
of the particle are shown at one instant in time. (a) Is the particle traveling clockwise
or counterclockwise (choose one) around the circle? (b) Is the particle’s speed increasing,
constant, or decreasing (choose one)?
22. Compute fx (2, −1) for f (x, y) =
p
2x2 + y 2 .
23. Circle those variables in the following list that must be held constant while calculating
w = f (x, y, z, t).
∂w
if
∂z
(a) w
(b) x
(c) y
(d) z
(e) t
24. Using the level curves for f (x, y), indicate whether each derivative is less than, equal to, or
greater than 0. Use the answer boxes.
0 1
2
3
fx (P )
0
fy (P )
0
fyy (P )
0
y
4
P
5
x
fxx (P )
0
fxy (P )
0
25. If w = f (u) and u is a function of x and y then
∂2w
∂y 2
equals (circle one)
2
(a) (f 0 (u) ∂u
∂y ) .
2
∂ u
2
0
(b) f 00 (u)( ∂u
∂y ) + f (u) ∂y 2 .
(c) f 00 (u) ∂u
∂y .
2
(d) f 0 (u) ∂∂yu2 .
26. Give parametric equations for the curve formed by the intersection of the cylinder x2 + y 2 = 9
and the hyperbolic paraboloid z = xy.
27. The upper end of 4-meter-long ladder slides along a vertical wall as the bottom is pulled away
from the wall. If θ is the angle between the ladder and the horizontal, as shown, the position
vector r(θ) of the midpoint P of the ladder is (circle one)
y
a. h2 cos θ, 2 sin θi.
b. h2 sin θ, 2 cos θi.
c. h2 sin θ, 2 − 2 sin θi.
P
d. h2 cos θ, 2 − 2 cos θi.
e. h4 − 2 cos θ, 2 sin θi.
)
r(Θ
Θ
x
28. The function r(t) traces a space curve in R3 . If kr(t)k = 4 then (circle all that must be true)
a. the space curve is a circle of radius 2.
b. the space curve is a circle of radius 4.
c. the space curve lies on a sphere of radius 2.
d. the space curve lies on a sphere of radius 4.
29. The function r(t) traces a curving path in R3 . If kr(t)k = 1 then (circle all that apply)
a. r0 (t) is orthogonal to r(t) for all t.
b. r0 (t) is parallel to r(t) for all t.
c. r0 (t) = 0 for all t.
d. r0 (t) is neither orthogonal to nor parallel to r(t) for all t.
30. Assuming f (2, 3) = 4, fx (2, 3) = −1, and fy (2, 3) = 5 estimate f (1.9, 3.1).
x2 − y 2
or show the limit does not exist.
(x,y)→(1,−1) x + y
31. Calculate
lim
32. Find r(t) if r0 (t) = et , 2 sin(2t) and r(0) = h2, 1i. Show your work.
33. Find the point on the graph of z = x2 + xy at which the tangent plane is parallel to the plane
2x − 2y + z = 10. Show your work.
34. Suppose f (x, y) = exy . Then fxx equals (circle one):
a. exy .
b. ey .
c. y 2 exy .
d. y 2 ex .
e. exy + y 2 exy .
35. Using the level curves for f (x, y), indicate whether each derivative is less than, equal to, or
greater than 0. Use the answer boxes.
f=4
f=2
y
P
fx (P )
0
fxx (P )
0
fxy (P )
0
fy (P )
0
fyy (P )
0
f=
0
x
36. Give a trigonometric parametrization for the ellipse that is the intersection between the
cylinder x2 + y 2 = 9 and the hyperbolic paraboloid z = y 2 − x2 .
37. A particle’s position is given by r(t), a twice differentiable vector function where r(t) · r(t) =
1 + t2 . Differentiate the equation to show to show r(t) · a(t) is a function of kv(t)k. Show
your work.
38. The derivative
a.
b.
y
∂
∂y y−3x
equals (circle one)
1
1−3x .
3x
.
(y−3x)2
3x
c. − (y−3x)
2.
d. − 21 .
e. 1.
39. A particle travels counterclockwise around a circle of radius 2. The velocity and acceleration
vectors are shown at t = 1. Use an appropriate symbol (<, =, or >) to complete each of the
following relationships. If insufficient information is available leave the box empty.
aT (1)
a
kT(1)k
0
aN (1)
kN(1)k
2
0
2
v
40. Sketch the traces of the function z = f (x, y) = 4 − 43 x − 2y in each of the 3 coordinate planes.
Give the coordinates of the x, y, or z-intercepts of the traces and the equation for each trace.
z
y
x
41. This graph shows level curves for a differentiable function f (x, y); note that f (P ) = 1 and
f (Q) = 2. Answer the questions that follow, using the graph to (briefly) explain your choices.
2
f=
3
f=
0
1
f=
f=
1
f=
y
Q
P
2
f=
x
(a) Is fx (P ) greater than, less than or equal to 0? Explain.
(b) Is fy (P ) greater than, less than or equal to 0? Explain.
(c) Is fx (P ) greater than, less than or equal to fx (Q)? Explain.
(d) Is fy (P ) greater than, less than or equal to fy (Q)? Explain.
42. Find all second partial derivatives of f (x, y) = ey + y sin x. Show how each of the derivatives
is calculated and use proper notation.
43. Consider a particle moving along a space curve described by r(t).
√ At t = 2, r(2) =√h0, 3, 4i,
1
1
0
√
T(2) = 3 h2, 2, −1i, N(2) = 5 h0, 1, 2i, T (2) = h0, 2, 4i, κ(2) = 5 and ka(2)k = 4 5.
If you can, calculate the following quantities. Otherwise, clearly state that there is insufficient
information to perform the calculation. Show your work.
(a) The velocity v(2).
(b) The normal component of acceleration aN (2).
(c) The tangential component of acceleration aT (2).
44. Using sin t and cos t, give a parameterization for the intersection of the circular cylinder
x2 + y 2 = 4 with the plane 3x + 2y + z = 6.
√
45. A particle moves along a space curve with a velocity v(t) = 3 cos(πt), 3 sin(πt), 2 t .
a. What distance (arc length) does the the particle travel between t = 0 and t = 4? Show
your work.
b. If the particle moves from point P = (1, 1, 1) at t = 0 to point Q at t = 4, what is the
−−→
displacement kP Qk of the particle? Show your work. Hint: the fundamental theorem
of calculus for vector-valued functions might be useful.
√
46. The position of a particle is given by r(t) = 2 3 cos t, 2 sin t , and the graph of r(t) is shown
here.
4
3
y
1
-4 -3
-2
-1
1
2
3
4
-1
-3
-4
x
(a) Calculate r(π/6), v(π/6), and a(π/6).
(b) Plot r(π/6), and then plot v(π/6), and a(π/6) so that their tails are at the head of
r(π/6). Label each vector.
(c) Does the particle travel around the ellipse in the clockwise or counterclockwise direction?
(d) Is the particle’s speed increasing, decreasing, or constant when t = π/6? Justify your
answer.
47. Show that N0 (t) is perpendicular to N(t). Remember that N(t) is a unit vector. Hint: use
the product rule for dot products.
48. Consider the surface S represented by the equation x2 + 3y 2 + 4z 2 = 20.
a. Find an equation for the tangent plane to S at the point (2, −2, 1).
b. The equation for S defines z implicitly as a function of x and y, z = f (x, y). Find the
linear approximation for f (x, y) at (2, −2). Leave your answer in the form
f (x, y) ≈ 10 + 20(x − 30) + 40(y − 50) (of course, you should use the correct numbers).
c. Use the linear approximation to estimate the number f (1.9, −1.2).
49. Suppose f (x, y) = 4 − (x − 2)2 − (y + 1)2 and x = r cos θ and y = r sin θ. Use the chain rule
to calculate fr and fθ when x = −3 and y = 4. Hint: express the derivatives as functions of
x and y.
50. Suppose g(x, y) = x2 − 2y 2 + 6 and x = 2 cos 2t and y = 2 sin 2t. Use the chain rule to
calculate g 0 (π/4).
51. Given f (x, y) = x3 y 2 , the equation for the tangent plane to z = f (x, y) at (a, b) = (1, 1) is
(circle one)
(a) z = 1 + 3x − 2y.
(b) z = 1 + 3(x − 1) + 2(y − 1).
(c) z = 2 + 3(x − 1) + 2(y − 1).
(d) z = 3(x − 1) + 2(y − 1).
2
8x+12
52. If f (x, y) = 2x+3
4y+1 then fx = 4y+1 and fy = − (4y+1)2 . Find the linear approximation of f (x, y)
at (0, 0) and use it to approximate the number f (0.1, −0.1). Show your work.
53. If w is a function of x, y, and z and x, y, and z are all functions of t then w0 (t) equals (circle
one)
dw dw dw
+
+
.
dx
dy
dz
∂w dx ∂w dy ∂w dz
(b)
+
+
.
∂x dt
∂y dt
∂z dt
∂w dx ∂w dy ∂w dz
(c)
.
∂x dt ∂y dt ∂z dt
∂w dx ∂w dy
(d)
+
.
∂x dy
∂y dz
(a)
54. Suppose f (x, y) is a function with derivatives fx = 2x and fy = 2y with x = r cos θ and
y = r sin θ. Use the chain rule to calculate fr when r = 2 and θ = 0. Show your work.
55. Given that z is defined implicitly as a function of x and y by the equation F (x, y, z) =
x3 z − y = 0, the partial derivative zx equals (choose one)
zy
.
xy
Fz
(b) − .
Fx
Fx
(c) − .
Fz
(d) none of the above.
(a) −
56. If z = x2 y + 3xy 4 , where x = sin(2st) and y = s cos(t), use the chain rule, not substitution,
to find zt when s = 1 and t = 0. Show your work.