9:02 MATH 122 WORKSHEET 4 lll 36% FEBRUARY 12, 2025 Solve the following problems on a separate sheet and fully justify your answers. Definition: We say that the function has: (i) a removable discontinuity at x=a if lim f(x) exists and is finite valued but lim f(x) = f (a) xa (ii) a jump discontinuity at x = a x-a if both lim f(x) and lim f (x) exist and are finite but lim f (x) = lim f(x). x-a x->a (iii) an infinite discontinuity at x=a x->>a* x-a if at least one of the one-sided limits: lim f (x) or lim f(x) is infinite valued. xa 1. The graph of a function y = f(x) is shown below. (a) At which values of x is f not continuous? D4-X Classify each point of discontinuity as removable, jump, or infinite. (b) At which values of x is f not differentiable? Why? (c) Determine the values of the left and right-sided derivatives at -2. f continuous at x=a lim f(x) = f(a) x-a f differentiable at x=a lim Ar-0 f(a+Ax)-f(a) Ar = m for some mER 2. Let f(x) = -1 y = f(x) │x if |x| 1 x²-1 0 if |x| = 1 left-side derivative at -2: f(-2+Ax)-f(-2) Ax f' (-2) = lim right-side derivative at -2: ƒ(−2 + Ax) − f(−2) f'(-2) = lim Ax-0 Ax (a) Sketch a graph of ƒ and clearly indicate its points of removable discontinuity. Hint: Observe that x2-1 (|x| -1 )( |x| +1). So, f(x)= 1 x+1 whenever |x| # 1. (b) How should f(x) be re-defined at each point of discontinuity in order to make it continuous there? Support your answers. 3. Let f(x)= -{ √ if 0 ≤ x ≤4 where c denotes a real constant. c(x-4)+2 if x > 4 (a) Show that lim f (x) = lim f (x) and that fe is continuous at x=4 for any constant c. x+4+ x-4 (b) Sketch the curves y = f(x) on separate graphs for c=0, c = 1, and c=-1. (c) Determine the unique value of c for which fe is differentiable at x=4, i.e., find c so that (f.)'+ (4) = (fc)' (4). |||
9:02 MATH 122 WORKSHEET 4 lll 36% FEBRUARY 12, 2025 Solve the following problems on a separate sheet and fully justify your answers. Definition: We say that the function has: (i) a removable discontinuity at x=a if lim f(x) exists and is finite valued but lim f(x) = f (a) xa (ii) a jump discontinuity at x = a x-a if both lim f(x) and lim f (x) exist and are finite but lim f (x) = lim f(x). x-a x->a (iii) an infinite discontinuity at x=a x->>a* x-a if at least one of the one-sided limits: lim f (x) or lim f(x) is infinite valued. xa 1. The graph of a function y = f(x) is shown below. (a) At which values of x is f not continuous? D4-X Classify each point of discontinuity as removable, jump, or infinite. (b) At which values of x is f not differentiable? Why? (c) Determine the values of the left and right-sided derivatives at -2. f continuous at x=a lim f(x) = f(a) x-a f differentiable at x=a lim Ar-0 f(a+Ax)-f(a) Ar = m for some mER 2. Let f(x) = -1 y = f(x) │x if |x| 1 x²-1 0 if |x| = 1 left-side derivative at -2: f(-2+Ax)-f(-2) Ax f' (-2) = lim right-side derivative at -2: ƒ(−2 + Ax) − f(−2) f'(-2) = lim Ax-0 Ax (a) Sketch a graph of ƒ and clearly indicate its points of removable discontinuity. Hint: Observe that x2-1 (|x| -1 )( |x| +1). So, f(x)= 1 x+1 whenever |x| # 1. (b) How should f(x) be re-defined at each point of discontinuity in order to make it continuous there? Support your answers. 3. Let f(x)= -{ √ if 0 ≤ x ≤4 where c denotes a real constant. c(x-4)+2 if x > 4 (a) Show that lim f (x) = lim f (x) and that fe is continuous at x=4 for any constant c. x+4+ x-4 (b) Sketch the curves y = f(x) on separate graphs for c=0, c = 1, and c=-1. (c) Determine the unique value of c for which fe is differentiable at x=4, i.e., find c so that (f.)'+ (4) = (fc)' (4). |||
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section: Chapter Questions
Problem 35RE
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Transcribed Image Text:9:02
MATH 122
WORKSHEET 4
lll 36%
FEBRUARY 12, 2025
Solve the following problems on a separate sheet and fully justify your answers.
Definition: We say that the function has:
(i) a removable discontinuity at x=a
if lim f(x) exists and is finite valued but lim f(x) = f (a)
xa
(ii) a jump discontinuity at x = a
x-a
if both lim f(x) and lim f (x) exist and are finite but lim f (x) = lim f(x).
x-a
x->a
(iii) an infinite discontinuity at x=a
x->>a*
x-a
if at least one of the one-sided limits: lim f (x) or lim f(x) is infinite valued.
xa
1. The graph of a function y = f(x) is shown below.
(a) At which values of x is f not continuous?
D4-X
Classify each point of discontinuity as removable, jump, or infinite.
(b) At which values of x is f not differentiable? Why?
(c) Determine the values of the left and right-sided derivatives at -2.
f continuous
at x=a
lim f(x) = f(a)
x-a
f differentiable
at x=a
lim
Ar-0
f(a+Ax)-f(a)
Ar
= m
for some mER
2. Let f(x) =
-1
y = f(x)
│x
if |x| 1
x²-1
0
if |x| = 1
left-side derivative at -2:
f(-2+Ax)-f(-2)
Ax
f' (-2) = lim
right-side derivative at -2:
ƒ(−2 + Ax) − f(−2)
f'(-2) = lim
Ax-0
Ax
(a) Sketch a graph of ƒ and clearly indicate its points of removable discontinuity.
Hint: Observe that x2-1 (|x| -1 )( |x| +1). So, f(x)=
1
x+1
whenever |x| # 1.
(b) How should f(x) be re-defined at each point of discontinuity in order to make it
continuous there? Support your answers.
3. Let f(x)=
-{
√
if 0 ≤ x ≤4
where c denotes a real constant.
c(x-4)+2 if x > 4
(a) Show that lim f (x) = lim f (x) and that fe is continuous at x=4 for any constant c.
x+4+
x-4
(b) Sketch the curves y = f(x) on separate graphs for c=0, c = 1, and c=-1.
(c) Determine the unique value of c for which fe is differentiable at x=4,
i.e., find c so that (f.)'+ (4) = (fc)' (4).
|||
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