section 2.8 - continuity. continuous pt. discontinuity at x = 0 inf. discontinuity at x = 1 pt....
TRANSCRIPT
2f x x 2x 1
continuous
2
1f x
x 1
continuous
2
xf x
x x
x
x x 1
pt. discontinuity at x = 0
inf. discontinuity at x = 1
2
x 3f x
x 9
x 3
x 3 x 3
pt. discontinuity at x = 3
inf. discontinuity at x = -3
2
2x 3 x 1f x
x x 1
2 1 3 1
21 1
continuous
1
x 1 x 2f x 2
3 x x 2
12 1 2
2
3 2 1
jump discontinuity at x = 2
Find the value of a which makes the function below continuous
3
2
x x 2f x
ax x 2
3
2
3
2
x x 2f x
ax x
2 8
2 4a2 a
4a 8 a 2
No Calculator
Find (a, b) which makes the function below continuous
2 x 1
f x ax b 1 x 3
2 x 3
As we approach x = -1 2 = -a + b
As we approach x = 3 -2 = 3a + b
4 4aa 1 b 1
1,1
No Calculator
tanxConsider the function f defined on x by f x
2 2 sinx
No Calculator
for all x . If f is continuous at x = , then f
A. 2 B. 1 C. 0 D. -1 E. -2
sinx
tanx cos xf xsinx sinx
sinx 1
cos x sinx 1 1
1 Dcos x cos
No Calculator
Which function is NOT continuous everywhere?
2 / 3
2
2
2
A. y x
B. y x
C. y x 1
xD. y
x 14x
E. yx 1
undefined at x = -1
Calculator Required
Which of the following is true about 2
2
x 1f x ?
2x 5x 3
I. f is continuous at x = 1 II. The graph of f has a vertical asymptote at x = 1III. The graph of f has a horizontal asymptote at y = 1/2
A. I B. II C. III D. II, III E. I, II, III
I. f(1) results in zero in denominator….NO
II. Since x – 1 results in 0/0, it is a HOLE, NOT asymptote
III. 2
2x
x 1lim
2x 5x 3
2
2x
x 2x 1lim
2x 5x 3
X X
X X
1
2
The graph of the derivative of a function f is shown below.Which of the following is true about the function f? I. f is increasing on the interval (-2, 1) II. f is continuous at x = 0III. f has an inflection point at x = -2
A. I B. II C. III D. II, III E. I, II, III
NOYESYES
Calculator Required
Calculator RequiredLet m and b be real numbers and let the function f be defined by:
21 3bx 2x x 1
f xmx b x 1
If f is both continuous and differentiable at x = 1, then:A. m 1, b 1
B. m 1, b 1
C. m 1, b 1
D. m 1, b 1
E. none of these
3b 4x x 1f ' x
m x 1
3b 4 m 1 3b 2 m b
3b m 4
2b m 3
If x = 1