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Math 151 WIR, Spring 2010, Benjamin
Aurispa
Math 151 Week in Review 6
Sections 3.5, 3.6, & 3.7
1. Differentiate the following functions.
(a) f (x) = (5x + cos 3x)−6
(b) f (x) =
sin2 4x
x6 − tan 5x
q
(c) f (x) =
3x +
p
8
3
− 7x
x2
!4
5x2 + sec 2x
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
(d) f (x) = cot2 (x2 − x) + csc(cos(tan 4x))
2. Given the following table of values, calculate the indicated derivatives.
x
f
f0
g
g0
0
2
1
1
−2
3
π
6
π
3
6
−3
2
1
5
2
0
(a) h0 (2) if h(x) = g(f (x))
(b) G0 (1) if G(x) = [f (4x − 3)]3
(c) F 0 (0) if F (x) = f (sin 2x + 1) + cos(g(x2 + 3x))
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
5−x
3. Find an equation of the tangent line to the graph of f (x) = √
at the point where x = 1.
x2 + 3
4. Find
dy
dx
for the equation 3y 4 − 2x2 y 2 = 7x5 + y
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
5. Find
dy
dx
for the equation cos(2x − y) + sin 3y = 4xy.
6. Find an equation of the tangent line to the graph of (x2 − 7y 3 )2 = 4y at the point (−3, 1).
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
7. Show that the curves x2 + y 2 = 2x and x2 + y 2 = 6y are orthogonal.
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
8. The motion of a particle is given by the vector function r(t) =< −5 cos t, 2 sin t >.
(a) Sketch the curve.
(b) What is the velocity of the object at the point (0, −2)?
(c) In what direction is the particle traveling?
9. Find a unit tangent vector to the curve r(t) =< t3 − 1, 3 − 3t2 > at the point (−2, 0).
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
10. Find vector and parametric equations for a tangent line to the curve r(t) =< 2t + cos t, 4 sin 2t > at
t = π.
11. The position of a thrown water balloon in feet after t seconds is given by r(t) =< 2t, 10t − 4t2 >.
(a) What are the velocity and speed of the balloon after 2 seconds?
(b) With what speed will the balloon hit the ground?
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
12. Find the angle of intersection of the curves r1 (t) =< t + 4, t2 − 9 > and r2 (s) =< 5 − s, s2 − 6 >.
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