Graph of a Curve

1
Graph of a Curve
Continuity
This curve is continuous
These curves are discontinuous
hole
gap
GAP
Smoothness
This curve is smooth
These curves are not smooth
corner
cusp
2
Graph of a Curve (contd)
Increasing
Decreasing
Constant
A function f is increasing on an open interval I
if, for any x1 and x2 in I, with x1 lt x2, we have
f(x1) lt f(x2).
A function f is decreasing on an open interval I
if, for any x1 and x2 in I, with x1 lt x2, we
have f(x1) gt f(x2).
A function f is constant on an open interval I
if, for any x1 and x2 in I, we have f(x1)
f(x2).
Local (or Relative) Extrema
f(x1) is a local max and also a global max
A function f has a local maximum at x x0 if
locally, f(x0) is greater than all the
surrounding values of f(x). We call this f(x0) a
local maximum of f. A function f has a local
minimum at x x0 if locally, f(x0) is less than
all the surrounding values of f(x). We call this
f(x0) a local minimum of f.
f(x3) is a local max
x1
x2
x3
Global (or Absolute) Extrema
A function f has a global maximum at x x0 if
f(x0) is greater than or equal to all values of
f(x). We call this f(x0) the global maximum of
f. A function f has a global minimum at x x0 if
f(x0) is less than or equal to all values of
f(x). We call this f(x0) the global minimum of f.
f(x2) is a local min
3
Use each graph to find the domain and range of
the function
Domain 5, 2) ? (1, 5
Domain 4, 4)
Range 3, 1) ? 2,4)
Range (2, 2
Domain (8, 3) ? 1, 4)
Domain (?, 5) ? (5, ?)
Range (?, 2) ? (1, 3
Range 3 ? 2, 1) ? (1, ?)
4
Use the graph of the function f given below to
answer the following questions
1) Find f(5), f(2), and f(6) f(5) 3, f(2)
1, f(6) 1
12) List the interval(s) on which f is
decreasing? (9, 2) ? (0, 1) ? (3, 4)
2) Is f(1) positive? No f(5)? Yes
13) List the interval(s) on which f is
increasing? (2, 0) ? (1, 3) ? (4, 5)
3) Is f(2) negative? No f(4)? Yes
14) List the interval(s) on which f constant? (5,
6)
4) What is the domain of f ? (9,6
15) If any, list all the local maxima? 1, 4 At
which x values? 0, 3
5) What is the range of f ? 3,4
6) What are the x-intercepts? 7.5, 1, 1
16) If any, list all the local minima? 3, 0, 1
At which x values? 2, 1, 4
7) What is the y-intercept? 1
8) Is f continuous? Yes
17) Is there any global maximum? Yes, 4 global
minimum? Yes, 3
9) Is f smooth? No
10) For what value(s) of x does f(x) 0? 7.5,
1, 1 f(x) 1? 0, 2, 4
11) How often does the line y 1 intersect the
graph? 3 times y ½? 4 times y 2? 2 times