Continuities
What are the three conditions that must be true for a function to be continuous at x=a
What type of discontinuity occurs when the limit exists, but the function has a missing point?
Removable Discontinuity
Prompt: Can the IVT be used to prove that the continuous function
f(x)=x3-6x+2
has a zero on [0,1]?
Yes
Explanation:
f(0)=2
f(1)=-3
If limx→4f(x)=∞
What is the vertical asymptote?
x=4
What was our groups name??
The spongebob squareroots!
Is f(x)=x2−9/x−3 continuous at x=3
No. There is a removable discontinuity (hole) at x=3
Identify the type of discontinuity at x=2
f(x)=x−2/x+3
Infinite discontinuity/VA
Prompt: A function f is continuous on [2,7], with
with f(2)=10 and f(7)=4
Can the IVT guarantee that f(c)=7 for some value c from the interval [2,7]
Yes; since 7 lies between 10 and 4, the IVT guarantees that at some point on the graph, f(c) will equal 7
What are the vertical asymptotes for this function:
x+3/x2-16
x=4 and -4
Which spongebob character was dhriti in our group?
Patrick!
Find k so that the function is continuous at x=2
f(x)=x2+k x<2
3x+2 x≥2
k=4
What type of discontinuity occurs when the left-hand and right-hand limits are different?
Jump Discontinuity
Prompt: A function f is continuous on [2,6] with f(2)=3 and f(6)=9
A student claims the IVT guarantees that f(c)=5 for EXACTLY ONE value of c between 2 and 6; is this claim correct?
NO! The IVT guarantees at LEAST one value of c such that f(c)=5; the function could technically reach 5 multiple times
Prompt: Find the vertical asymptote of
f(x)=x+2/x2-9
x=3 and x=-3
What instrument does squidward play?
The Clarinet
a function has f(2)=5 and the limx→2f(x)=3
Is the function continuous at x=2?
NO!! The limit does NOT equal the function's value
Identify the discontinuity in
f(x)=x2-4/x-2
Removable discontinuity at x=2
The function f is continuous on [2,6], with
f(2)=−3 and f(6)=5
Which statement is guaranteed by the Intermediate Value Theorem?
A. f(x)=1 exactly once between 2 and 6
B. f(x)=0 at least once between 2 and 6
C. f(x) is increasing between 2 and 6
D. f(x) has a maximum between 2 and 6
Answer: B. f(x)=0 at least once between 2 and 6
Determine the Horizontal Asymptote:
f(x)=4x3-2x+1/2x3+7
y=2; degrees are equal, so you use the ratio of the leading coefficients!
What is Mr.Krabs first name?
Eugene
Find a so that the function is continuous everywhere
f(x)=ax+4 when x<1 and x2+a when x≥1
this is impossible! there is NO VALUE of a that makes the function continuous everywhere!!
Let
f(x)=x2-4/x2-x-2
At which value of x does f have a removable discontinuity?
x=2
Prompt: The function ff is continuous on [1,5]. If
f(1)=−4 and f(5)=6
which of the following must be true?
A. f(x)=0 exactly once on (1,5)
B. f(x)=0 at least once on (1,5)
C. f(x) is increasing on [1,5]
D. f(x) has a local maximum on (1,5)
Answer: B. f(x)=0 at least once on (1,5)
Consider this function:
3x2+2x-1/x2-1
What are the horizontal and vertical asymptotes?
vertical asymptote: x=1
horizontal asymptote: y=3
GARY!!