Math 151 Week in Review
Monday Oct 11, 2010
Instructor: Jenn Whitfield
Thanks to Amy Austin for contributing some problems.
All prolbems in this set are copywrited
Section 3.5
The Chain Rule:
If n is any real number and u = g(x) is
d n
du
differentiable then
(u ) = nun−1
or
dx
dx
d
[g(x)]n = n[g(x)]n−1 • g′ (x).
dx
Section 3.6
6. Find
dy
for cos(2x) − sin(x + y) = 1.
dx
7. Find the equation of the line tangent to
x2 + y 2 = 2 at (1,1).
8. Find
dy
if x4 − 4x2 y 2 + y 3 = 0.
dx
9. Regard y as the independent variable and
x as the dependent variable. Use implicit
dx
given the equation
differentiation to find
dy
(x2 + y 2 )2 = ax2 y.
1. Given h = f ◦ g, g(3) = 6, g′ (3) = 4, f ′ (3) =
2, and f ′ (6) = 7. Find h′ (3).
10. If x[f (x)]3 + xf (x) = 6 and f (3) = 1, find
f ′ (3).
2. Differentiate the following functions
11. Show that the curves 2x2 + y 2 = 3 and x =
y 2 are orthogonal.
(a) f (x) = (4 − 3x2 )4
√
(b) f (x) = x cos( x)
(c) f (t) =
sin(2t3
(d) f (x) =
r
+ 4t − 1)
3x +
(e) h(x) = √
Section 3.7
q
1
x
+ x2
2
x3
+5
2
(f) f (x) = tan (4x4 − 5)
(g) y = (x2 + 1)4 (6 − 2x)3
3. Find the equation of the
tangent line to
π
,1 .
sin x + cos(2x) at
6
4. If f and g are functions whose graphs are
shown below, let u(x) = f (g(x)), v(x) =
g(f (x)) and w(x) = g(g(x)). Find the following:
(a) u′ (1)
(b) v ′ (1)
(c) w′ (1)
5. Suppose f is differentiable on (−∞, ∞) and
α is a real number. Let G(x) = [f (x)]α and
F (x) = f (xα ). Find expressions for F ′ (x)
and G′ (x).
12. Find the angle between the tangent vector
and the position vector for
r(t) = ht2 , 2t3 i at the point where t = −1.
13. Find the vector and parametric equations of
the line tangent to r(t) = ht3 + 2t, 4t − 5i at
the point where t = 2.
14. Find the unit tangent vector to the curve
r(t) = ht2 , 3t3 i at the point (1, −3).
15. Find the angle of intersection of the curves
r1 (s) = hs − 2, s2 i and r2 (t) = h1 − t, 3 + t2 i.