MATH 151 Engineering Math I, Spring 2014
JD Kim
Week6 Section 3.4, 3.5, 3.6, 3.7
Section 3.4 Derivatives of Trigonometric Functions
Two Special limits
1.
sin x
=1
x→0 x
lim
2.
cos x − 1
=0
x→0
x
lim
Ex1) Find the limit
1-1) limx→0
sin x
3x
1
1-2) limx→0
sin 9x
7x
1-3) limx→0
sin 8x
sin 7x
1-4) limx→0
tan2 4x
x2
1-5) limx→0
cos x − 1
sin x
2
Derivatives of Trigonometric Functions
1. (sin x)′ = cos x
2. (cos x)′ = − sin x
3. (tan x)′ = sec2 x
4. (sec x)′ = sec x tan x
5. (cot x)′ = − csc2 x
6. (csc x)′ = − csc x cot x
Ex2) Find the derivative.
√
2-1) f (x) = sec2 x + 4 tan x + x x
2-2) g(t) =
2 cos t + 1
cos t + t
3
Ex3) Find the equation of the tangent line to the graph of f (x) = 2 sin x at
π
x= .
3
4
Ex4) For what value(s) of x does the graph of f (x) = x + 2 sin x have a horizontal
tangent?
5
Section 3.5 Chain Rule
The Chain Rule If the derivatives g ′ (x) and f ′ (g(x)) both exist, and F = f ◦ g
is the composite function defined by F (x) = f (g(x)), then F ′ (x) exists and is given
by the product
F ′ (x) = f ′ (g(x))g ′(x)
In Leibniz notation, if y = f (u) and u = g(x) are both differentiable functions, then
dy du
dy
=
·
dx
du dx
Ex5) Find the derivative
5-1) f (x) = sin(2x) + cos(5x2 )
5-2) g(x) = tan(cos(x))
5-3) h(x) = sec(cos(sin(4x2 )))
6
The Power Rule combined with the Chain Rule (Generalized Power
Rule)
If n is any real number and u = g(x) is differentiable, then
du
d n
(u ) = n · un−1 ·
dx
dx
Alternatively,
d
[g(x)]n = n · [g(x)]n−1 · g ′(x).
dx
Ex6) Find the derivative
6-1) f (x) =
1
(x2 + 5x + 4)10
√
6-2) g(x) = x3 ( x + 5)3
6-3) h(x) = sin(3x) + sin3 x
6-4) f (x) =
p
cos(sin2 x)
7
6-5) g(x) =
q
p
√
x+ x+ x
√
Ex7) Find the equation of the tangent line to the graph of f (x) = 8 4 + 3x at
x = 4.
Ex8) Suppose w = u ◦ v and u(0) = 1, v(0) = 2, u′ (0) = 3, u′(2) = 4, v ′ (0) = 5
and v ′ (2) = 6. Find w ′ (0).
Ex9) If F (x) = f (cos x), G(x) = cos(f (x)) and H(x) = [f (sin x)]3 , find F ′ (x), G′ (x)
and H ′ (x).
8
Ex10) Find the all points on the curve y = sin(2x) + cos(2x) where the tangent
line is horizontal.
9
Section 3.6 Implicit Differentiation
Why we differentiate Implicitly?
Suppose y = f (x). In this case, we say y is an explicit function of x and we can
dy
= f ′ (x). In this section, we investigate how to
therefore differentiate as usual:
dx
differentiate if y cannot be written as an explicit function of x, that is y is implicitly
defined as a function of x.
Ex11) Find
dy
for the following equations.
dx
11-1) y 3 = 2x2 + y 4
11-2) 2y 2 + xy = x2 + 3
10
11-3) x sin y + cos(2y) = cos y
11-4) (x2 + y 2)3 = 2y 4 + 6x2
Ex12) Find the tangent line to the ellipse
11
√
x2 y 2
+
= 1 at the point (−1, 4 2).
9
36
Definition Orthogonal Curves
We say two curves are orthogonal if their tangent lines are perpendicular at every
point of intersection.
Ex13) Prove the curves x2 − y 2 = 5 and 4x2 + 9y 2 = 72 are orthogonal.
12
Ex14) If (g(x))2 + 12x = x2 g(x), and g(3) = 4, find g ′ (3).
Ex15) Regard y as the independent variable and x as the dependent variable, and
dx
for the equation (x2 + y 2 )2 = 2x2 y.
use implicit differentiation to find
dy
13
Section3.7 Derivatives of Vector Functions
Definition The derivative of a vector function r at a number a, denoted by r ′ (a)
is
r ′ (a) = lim
h→0
r(a + h) − r(a)
h
if this limit exists, and
r(t) − r(a)
t→a
t−a
If r(t) =< x(t), y(t) > is a vector function, then
r ′ (a) = lim
r ′ (t) =< x′ (t), y ′(t) >= x′ (t)ı + y ′ (t)
if both x′ (t) and y ′(t) exist.
Definition Velocity and speed.
If r(t) =< x(t), y(t) > is a vector function representing the position of a particle
at time t, then
′
′
Instantaneous velocity at time t is r ′ (t) =<
p x (t), y (y) >.
′
′
2
Instantaneous speed at time t is |r (t)| = (x (t)) + (y ′(t))2
Ex16) Sketch the following vector equations. Include the direction of the curve.
16-1) r(t) =< t, t2 >
14
16-2) r(t) =< 2 cos t, 3 sin t >
Ex17) If r(t) =< t2 − 4,
√
9 − t >, find the domain of r(t) and r ′ (t).
Ex18) r(t) =< t, 25t − 5t2 > is the position of a moving object at time t, where
position is measured in feet and time in seconds,
18-1) Find the velocity and speed at time t = 1.
18-2) With what speed does the object strike the ground?
15
Ex19) Find the unit tangent vector at t = 1 to the curve given by r(t) =<
t , 3t3 >.
2
Ex20) Find the angle of intersection of the curves r(t) =< 1 − t, 3 + t2 > and
s(u) =< u − 2, u2 >.
16