do now: use composite of continuous functions thm to show f(x) is continuous
TRANSCRIPT
DO NOW:Use Composite of Continuous Functions
THM to show f(x) is continuous.
xxf 1cos)(
22
sin2sinlim x
xxx
x
3
13 3
lim
x
xx
x
HW: Pg. 92-93 #1-3, 9-12, 16, 18, 23
2.4 – Rates of Change and Tangent Lines
TangentsDEFINITION
The tangent line to the curve y = f(x) at the point P(a, f(a)) is the line through P with slope
Provided that this limit exists.
ax
afxfm
ax
)()(lim
Example 1Find an equation of the tangent line to the
parabola y = x2 at the point P(1,1).
Tangent Line (2 equivalent statements)
(SLOPE OF A CURVE AT A POINT (a,f(a)) )
ax
afxfmPQ
)()(h
afhafmPQ
)()(
NORMAL TO A CURVEThe normal line to a curve at a point is the
line perpendicular to the tangent at that point.
Example 1 - ExtendedFind an equation for the normal line to the
parabola y = x2 at the point P(1,1).
Example 2Find an equation of the tangent line to the
hyperbola y = 3/x at the point (3,1).
Example 2 (Solution)
Average Velocity• Average velocity =
• The function f that describes the motion is called the position function of the object.
h
afhaf
time
ntdisplaceme )()(
Instantaneous VelocityNow suppose we compute the average
velocities over shorter and shorter intervals [a, a+h]
That is -> we let h approach 0.
The instantaneous velocity v(a) at time t = a to be the limit of the average velocities: h
afhafav
h
)()()( lim
0
Average vs Instantaneous VelocityAverage Velocity (secant line):
Average velocity =
Instantaneous Velocity (tangent line):
h
afhaf
time
ntdisplaceme )()(
h
afhafav
h
)()()( lim
0
Example 3Consider a ball dropped from a height of
450 m. Find:The velocity of the ball after 5 secondsHow fast the ball is traveling when it hits the
ground
Avg vs Instantaneous Rate of Change The average rate of change of y with
respect to x over the interval [x1, x2]:
The instantaneous rate of change of y with respect to x at x = x1
12
12 )()(
xx
xfxf
x
y
12
12
0
)()(limlim
12xx
xfxf
x
y
xxx
Example 4• Temperature readings T
(in C) were recorded every hour starting at midnight on a day in Whitefish, Montana.
• The time x is measured in hours from midnight.
• Find the average rate of change of temperature with respect to time– From noon to 3 PM– From noon to 2 PM– From noon to 1 PMEstimate the
instantaneous rate of change at noon.
Example 4 (solution)• Find the average rate of change of
temperature with respect to time– From noon to 3 PM– From noon to 2 PM– From noon to 1 PMEstimate the instantaneous rate of change at
noon.