ISC3701 ASSIGNMENT 4 PORTFOLIO (COMPLETE ANSWERS) 2024 (543545)- DUE 12 SEPTEMBER 2024
At f'(2) the upper piece is 4 and lower piece is 7 so f(x) is not differentiable everywhere.
Since the slopes of the function on the left and right are both positive the function
cannot have a local minimum or maximum at x= 2.
Only I is true.
For the function f(x) = (ax^3-6x), if x ≤ 1, & f(x) = (bx^2+4), x > 1 to be continuous and
differentiable, a = ..... - CORRECT ANSWERS-for the function to be continuous f(1) has
to equal f(1):
a(1^3) -6(1) = b(1^2) +4
a -6 = b +4
b=a-10
for the functions to be differentiable f'(1) has to equal f'(1):
3a(1^2) -6 = 2b(1)
3a -6 = 2b
plug b from the first equation in to find a:
3a -6 = 2(a -10)
a = -14
Find k if f(x) = (k) at x = 4 and f(x) = ((x^2 -16)/(x-4)) - CORRECT ANSWERS-1. f(4)
exists and is equal to 8
2. lim from the left and right are both 8
3. lim f(x) as x approaches 4 is 8 which equals f(4)
Find the midpoint Riemann Sum of cos(x^2) with n = 4, from [0, 2] - CORRECT
ANSWERS-Mid S4 = (1)(1/2)[cos(.25^2) + cos(.75^2) + cos(1.25^2) + cos(1.75^2)]
Mid S4 = (1)(1/2)[cos(.625) + cos(.5625) + cos(1.5625) cos(3.0625)]
Mid S4 = .824
If the function f is continuous for all real numbers and if f(x) = (x^2-7x +12)/(x -4) when x
≠ 4 then f(4) = - CORRECT ANSWERS-Factor numerator so
f(x) = (x-3)(x-4)/(x-4) = x-3
f(4)=4-3
f(4) = 1
If f(x) = (x^2+5) if x < 2, & f(x) = (7x -5) if x ≥ 2 for all real numbers x, which of the
following must be true?
I. f(x) is continuous everywhere.
II. f(x) is differentiable everywhere.
III. f(x) has a local minimum at x = 2. - CORRECT ANSWERS-At f(2) both the upper and
lower piece of the discontinuity is 9 so the function is continuous everywhere.
, k must equal 8
If f(x) is continuous and differentiable and f(x) = (ax^4 +5x) for x ≤ 2, & f(x)= (bx^2 -3) for
x > 2 , then b =... - CORRECT ANSWERS-Plug x = 2 into both pieces.
f(x) = (16a +10) for x ≤ 2, & (4b -6) for x > 2
They must be equal to be continuous
16a +10 = 4b -6
a=.25b-1
Take the derivative of both pieces of this function and plug in x = 2
f(x) = (32a +5) for x ≤ 2, & f(x) = (4b -3) for x > 2
They must be equal to be differentiable
32a +5 = 4b -3
plug in the first equation to find b
32(.25b-1)+5= 4b-3
b=6
If f is continuous for a ≤ x ≤ b, then at any point x = c, where a < c < b, which of the
following must be true?
a. f(c) = (f(b) - f(a))/(b-a)
b. f(a) = f(b)
c. f(c) = 0
d. lim f(x) as x approaches c = f(c) - CORRECT ANSWERS-In order for f(x) to be
continuous at point c, there are three conditions that need to be fulfilled:
1. f(c) exists
2. lim f(x) as x approaches c exists
3. lim f(x) as x approaches c = f(c)
Find the midpoint Riemann Sum of cos(x^2) with n = 4, from [0, 2] - CORRECT
ANSWERS-Mid S4 = (1)(1/2)[cos(.25^2) + cos(.75^2) + cos(1.25^2) + cos(1.75^2)]
Mid S4 = (1)(1/2)[cos(.625) + cos(.5625) + cos(1.5625) cos(3.0625)]
Mid S4 = .824
If the function f is continuous for all real numbers and if f(x) = (x^2-7x +12)/(x -4) when x
≠ 4 then f(4) = - CORRECT ANSWERS-Factor numerator so
f(x) = (x-3)(x-4)/(x-4) = x-3
f(4)=4-3
f(4) = 1
If f(x) = (x^2+5) if x < 2, & f(x) = (7x -5) if x ≥ 2 for all real numbers x, which of the
following must be true?
I. f(x) is continuous everywhere.
II. f(x) is differentiable everywhere.
III. f(x) has a local minimum at x = 2. - CORRECT ANSWERS-At f(2) both the upper and
lower piece of the discontinuity is 9 so the function is continuous everywhere.
At f'(2) the upper piece is 4 and lower piece is 7 so f(x) is not differentiable everywhere.
Since the slopes of the function on the left and right are both positive the function
cannot have a local minimum or maximum at x= 2.
Only I is true.
For the function f(x) = (ax^3-6x), if x ≤ 1, & f(x) = (bx^2+4), x > 1 to be continuous and
differentiable, a = ..... - CORRECT ANSWERS-for the function to be continuous f(1) has
to equal f(1):
a(1^3) -6(1) = b(1^2) +4
a -6 = b +4
b=a-10
for the functions to be differentiable f'(1) has to equal f'(1):
3a(1^2) -6 = 2b(1)
3a -6 = 2b
plug b from the first equation in to find a:
3a -6 = 2(a -10)
a = -14
Find k if f(x) = (k) at x = 4 and f(x) = ((x^2 -16)/(x-4)) - CORRECT ANSWERS-1. f(4)
exists and is equal to 8
2. lim from the left and right are both 8
3. lim f(x) as x approaches 4 is 8 which equals f(4)
Find the midpoint Riemann Sum of cos(x^2) with n = 4, from [0, 2] - CORRECT
ANSWERS-Mid S4 = (1)(1/2)[cos(.25^2) + cos(.75^2) + cos(1.25^2) + cos(1.75^2)]
Mid S4 = (1)(1/2)[cos(.625) + cos(.5625) + cos(1.5625) cos(3.0625)]
Mid S4 = .824
If the function f is continuous for all real numbers and if f(x) = (x^2-7x +12)/(x -4) when x
≠ 4 then f(4) = - CORRECT ANSWERS-Factor numerator so
f(x) = (x-3)(x-4)/(x-4) = x-3
f(4)=4-3
f(4) = 1
If f(x) = (x^2+5) if x < 2, & f(x) = (7x -5) if x ≥ 2 for all real numbers x, which of the
following must be true?
I. f(x) is continuous everywhere.
II. f(x) is differentiable everywhere.
III. f(x) has a local minimum at x = 2. - CORRECT ANSWERS-At f(2) both the upper and
lower piece of the discontinuity is 9 so the function is continuous everywhere.
, k must equal 8
If f(x) is continuous and differentiable and f(x) = (ax^4 +5x) for x ≤ 2, & f(x)= (bx^2 -3) for
x > 2 , then b =... - CORRECT ANSWERS-Plug x = 2 into both pieces.
f(x) = (16a +10) for x ≤ 2, & (4b -6) for x > 2
They must be equal to be continuous
16a +10 = 4b -6
a=.25b-1
Take the derivative of both pieces of this function and plug in x = 2
f(x) = (32a +5) for x ≤ 2, & f(x) = (4b -3) for x > 2
They must be equal to be differentiable
32a +5 = 4b -3
plug in the first equation to find b
32(.25b-1)+5= 4b-3
b=6
If f is continuous for a ≤ x ≤ b, then at any point x = c, where a < c < b, which of the
following must be true?
a. f(c) = (f(b) - f(a))/(b-a)
b. f(a) = f(b)
c. f(c) = 0
d. lim f(x) as x approaches c = f(c) - CORRECT ANSWERS-In order for f(x) to be
continuous at point c, there are three conditions that need to be fulfilled:
1. f(c) exists
2. lim f(x) as x approaches c exists
3. lim f(x) as x approaches c = f(c)
Find the midpoint Riemann Sum of cos(x^2) with n = 4, from [0, 2] - CORRECT
ANSWERS-Mid S4 = (1)(1/2)[cos(.25^2) + cos(.75^2) + cos(1.25^2) + cos(1.75^2)]
Mid S4 = (1)(1/2)[cos(.625) + cos(.5625) + cos(1.5625) cos(3.0625)]
Mid S4 = .824
If the function f is continuous for all real numbers and if f(x) = (x^2-7x +12)/(x -4) when x
≠ 4 then f(4) = - CORRECT ANSWERS-Factor numerator so
f(x) = (x-3)(x-4)/(x-4) = x-3
f(4)=4-3
f(4) = 1
If f(x) = (x^2+5) if x < 2, & f(x) = (7x -5) if x ≥ 2 for all real numbers x, which of the
following must be true?
I. f(x) is continuous everywhere.
II. f(x) is differentiable everywhere.
III. f(x) has a local minimum at x = 2. - CORRECT ANSWERS-At f(2) both the upper and
lower piece of the discontinuity is 9 so the function is continuous everywhere.
