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Math 151 WIR, Spring 2014, Benjamin
Aurispa
Math 151 Week in Review 6
Sections 3.4-3.7
1. Calculate the following limits.
(a) lim
sin 9x
x(cos x + 1)
(b) lim
(cos x − 1) sin 5x
x2
(c) lim
cot 3x
csc 4x
(d) lim
tan2 3x
6x2
x→0
x→0
x→0
x→0
1
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Math 151 WIR, Spring 2014, Benjamin
Aurispa
2. Find the tangent line to the graph of f (x) = tan x + 4 at x = π4 .
3. For what values of x does f (x) = sin x − cos x have a horizontal tangent line in the interval [0, 2π].
4. Differentiate the following functions.
(a) f (x) = sec x cot x + x csc x
(b) g(x) =
2x − cos x
tan x + sin x
2
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Math 151 WIR, Spring 2014, Benjamin
Aurispa
(c) f (x) = (x3 + 1)6 +
p
4x2 + 8 + sin(5x3 ) − tan 5x
1
x
+ cos 3x + 2
5
3x
(d) f (x) =
(e) f (x) =
√
3
3
+ sec 2x
4x + 1
−6
(f) f (x) = sin2 4x + cot4 (x2 )
(g) f (x) = csc(cos 4x)
3
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Math 151 WIR, Spring 2014, Benjamin
Aurispa
5. Given the following table of values, calculate the indicated derivatives.
x
f (x)
f ′ (x)
g(x)
g ′ (x)
0
2
1
1
−2
3
π
6
π
3
6
−3
2
1
5
2
0
(a) h′ (2) if h(x) = g(f (x))
(b) G′ (1) if G(x) = [f (4x − 4)]3
(c) Find F ′ (x) if F (x) = cos(g(x2 + 3x))
5−x
6. Find an equation of the tangent line to the graph of f (x) = √
at the point where x = 1.
x2 + 3
4
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Math 151 WIR, Spring 2014, Benjamin
Aurispa
7. Find
dy
dx
for the equation 3y 4 − 2x2 y 2 + 7x5 − y = 18
8. Find
dy
dx
for the equation
9. Find
dx
dy
for the equation x4 (x2 + 7y 3 ) =
√
y cos 2x + sin 3y = 4xy.
4
y
5
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Math 151 WIR, Spring 2014, Benjamin
Aurispa
10. Show that the curves x2 + y 2 = 2x and x2 + y 2 = 6y are orthogonal at the point
slopes of the tangent lines to each curve at this point.
9 3
5, 5
11. Find a unit tangent vector to the curve r(t) =< t3 − 1, 3 − 3t2 > at the point (−2, 0).
6
by finding the
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Math 151 WIR, Spring 2014, Benjamin
Aurispa
12. Find a vector equation of a tangent line to the curve r(t) =< 2t + cos t, 4 sin 2t > at t = π.
13. 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?
7
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Math 151 WIR, Spring 2014, Benjamin
Aurispa
14. The curves r1 (t) =< t + 4, t2 − 9 > and r2 (s) =< 5 − s, s2 − 6 > intersect at the point (6, −5). Find
the angle of intersection at this point.
8