Q. The function f(x) = x^2 - 2x + 1 is differentiable at all points?
-
A.
True
-
B.
False
-
C.
Only at x = 0
-
D.
Only for x > 0
Solution
f(x) is a polynomial function, which is differentiable everywhere.
Correct Answer: A — True
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Q. The function f(x) = x^2 - 2x + 1 is differentiable at x = 2?
-
A.
Yes
-
B.
No
-
C.
Only left
-
D.
Only right
Solution
f(x) is a polynomial function, hence it is differentiable everywhere including at x = 2.
Correct Answer: A — Yes
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Q. The function f(x) = x^2 - 4 is:
-
A.
Always increasing
-
B.
Always decreasing
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C.
Neither increasing nor decreasing
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D.
Both increasing and decreasing
Solution
The function has a minimum at x = 0, hence it is neither always increasing nor decreasing.
Correct Answer: C — Neither increasing nor decreasing
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Q. The function f(x) = x^2 - 4x + 4 can be expressed in which form?
-
A.
(x - 2)^2
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B.
(x + 2)^2
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C.
(x - 4)^2
-
D.
(x + 4)^2
Solution
f(x) = (x - 2)^2 is the completed square form.
Correct Answer: A — (x - 2)^2
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Q. The function f(x) = x^2 - 4x + 4 is differentiable at x = 2?
-
A.
Yes
-
B.
No
-
C.
Only left
-
D.
Only right
Solution
f(x) is a polynomial function, hence differentiable everywhere including at x = 2.
Correct Answer: A — Yes
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Q. The function f(x) = x^2 - 4x + 4 is differentiable everywhere?
-
A.
True
-
B.
False
-
C.
Only at x = 0
-
D.
Only at x = 2
Solution
f(x) is a polynomial function, hence it is differentiable everywhere.
Correct Answer: A — True
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Q. The function f(x) = x^2 for x < 1 and f(x) = 2x - 1 for x ≥ 1 is differentiable at x = 1?
-
A.
Yes
-
B.
No
-
C.
Only continuous
-
D.
Only from the left
Solution
f'(1) from left = 2 and from right = 2; hence, f is continuous but not differentiable at x = 1.
Correct Answer: B — No
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Q. The function f(x) = x^2 sin(1/x) for x ≠ 0 and f(0) = 0 is differentiable at x = 0. True or False?
-
A.
True
-
B.
False
-
C.
Depends on x
-
D.
Not enough information
Solution
True, as the limit of f'(x) as x approaches 0 exists and equals 0.
Correct Answer: A — True
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Q. The function f(x) = x^3 - 3x + 2 is differentiable at x = 1?
-
A.
Yes
-
B.
No
-
C.
Only left
-
D.
Only right
Solution
f(x) is a polynomial function, hence it is differentiable everywhere including at x = 1.
Correct Answer: A — Yes
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Q. The function f(x) = x^3 - 3x + 2 is differentiable everywhere. Find its critical points.
Solution
f'(x) = 3x^2 - 3 = 0 gives x = ±1, thus critical points are x = -1 and x = 1.
Correct Answer: B — 0
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Q. The function f(x) = x^3 - 3x + 2 is differentiable everywhere. What is f'(1)?
Solution
f'(x) = 3x^2 - 3, thus f'(1) = 0.
Correct Answer: A — 0
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Q. The function f(x) = x^3 - 6x^2 + 9x has how many local extrema?
Solution
Finding f'(x) = 3x^2 - 12x + 9. Setting f'(x) = 0 gives x = 1 and x = 3. Checking the second derivative shows one local maximum and one local minimum.
Correct Answer: B — 1
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Q. The function f(x) = { 1/x, x != 0; 0, x = 0 } is continuous at x = 0?
-
A.
Yes
-
B.
No
-
C.
Only from the right
-
D.
Only from the left
Solution
The limit as x approaches 0 does not equal f(0) = 0, hence it is not continuous at x = 0.
Correct Answer: B — No
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Q. The function f(x) = { 1/x, x ≠ 0; 0, x = 0 } is:
-
A.
Continuous at x = 0
-
B.
Not continuous at x = 0
-
C.
Continuous everywhere
-
D.
None of the above
Solution
The function is not continuous at x = 0 since the limit does not equal f(0).
Correct Answer: B — Not continuous at x = 0
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Q. The function f(x) = { 2x + 3, x < 1; x^2 + 1, x >= 1 } is continuous at x = ?
Solution
To check continuity at x = 1, we find the left limit (5) and the right limit (2). They are not equal, hence f(x) is not continuous at x = 1.
Correct Answer: B — 1
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Q. The function f(x) = { 3x + 1, x < 1; 2, x = 1; x^2, x > 1 } is continuous at x = 1 if which condition holds?
-
A.
3 = 2
-
B.
1 = 2
-
C.
2 = 1
-
D.
2 = 4
Solution
For continuity at x = 1, the left limit (3) must equal f(1) (2), which is not true.
Correct Answer: A — 3 = 2
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Q. The function f(x) = { 3x + 1, x < 1; 2x + 3, x >= 1 } is continuous at x = 1 if:
Solution
For continuity at x = 1, both pieces must equal 4, hence the function is continuous.
Correct Answer: A — 3
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Q. The function f(x) = { x + 2, x < 1; 3, x = 1; x^2, x > 1 } is continuous at x = ?
Solution
To check continuity at x = 1, we find the left limit (3) and the right limit (3). Both equal 3, hence f(x) is continuous at x = 1.
Correct Answer: B — 1
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Q. The function f(x) = { x^2, x < 0; 1, x = 0; x + 1, x > 0 } is continuous at x = 0?
-
A.
Yes
-
B.
No
-
C.
Only from the right
-
D.
Only from the left
Solution
Limit as x approaches 0 from left is 0, and f(0) = 1, hence it is not continuous at x = 0.
Correct Answer: A — Yes
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Q. The function f(x) = { x^2, x < 0; 2x + 1, x >= 0 } is continuous at which point?
-
A.
x = -1
-
B.
x = 0
-
C.
x = 1
-
D.
x = 2
Solution
To check continuity at x = 0, we find f(0) = 1 and limit as x approaches 0 is also 1.
Correct Answer: B — x = 0
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Q. The function f(x) = { x^2, x < 1; 2, x = 1; x + 1, x > 1 } is:
-
A.
Continuous everywhere
-
B.
Continuous at x = 1
-
C.
Not continuous at x = 1
-
D.
Continuous for x < 1
Solution
The function is not continuous at x = 1 because the left-hand limit does not equal the function value.
Correct Answer: C — Not continuous at x = 1
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Q. The function f(x) = { x^2, x < 1; 2x - 1, x >= 1 } is continuous at which point?
-
A.
x = 0
-
B.
x = 1
-
C.
x = 2
-
D.
x = -1
Solution
To check continuity at x = 1, we find f(1) = 1, limit as x approaches 1 from left is 1, and from right is also 1.
Correct Answer: B — x = 1
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Q. The function f(x) = { x^2, x < 1; 2x - 1, x >= 1 } is continuous at x = ?
Solution
To check continuity at x = 1, we find the limit from both sides. Both limits equal 1, hence f(x) is continuous at x = 1.
Correct Answer: B — 1
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Q. The function f(x) = { x^2, x < 1; 2x - 1, x ≥ 1 } is differentiable at x = 1 if which condition holds?
-
A.
f(1) = 1
-
B.
f'(1) = 1
-
C.
f'(1) = 2
-
D.
f(1) = 2
Solution
For differentiability, the left and right derivatives must equal at x = 1, hence f'(1) = 1.
Correct Answer: B — f'(1) = 1
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Q. The function f(x) = { x^2, x < 2; 4, x = 2; 2x, x > 2 } is continuous at x = 2 if:
-
A.
f(2) = 4
-
B.
lim x->2 f(x) = 4
-
C.
Both a and b
-
D.
None of the above
Solution
Both conditions must hold true for continuity at x = 2.
Correct Answer: C — Both a and b
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Q. The function f(x) = { x^2, x < 2; k, x = 2; 3x - 4, x > 2 } is continuous at x = 2 for which value of k?
Solution
To be continuous at x = 2, k must equal f(2) = 2^2 = 4.
Correct Answer: C — 4
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Q. The function f(x) = |x - 3| is continuous at which of the following points?
-
A.
x = 1
-
B.
x = 2
-
C.
x = 3
-
D.
x = 4
Solution
The function |x - 3| is continuous everywhere, including at x = 3.
Correct Answer: C — x = 3
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Q. The function f(x) = |x| is differentiable at x = 0?
-
A.
Yes
-
B.
No
-
C.
Only from the right
-
D.
Only from the left
Solution
f(x) = |x| is not differentiable at x = 0 because the left-hand and right-hand derivatives do not match.
Correct Answer: B — No
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Q. The general form of the family of curves for circles is given by:
-
A.
(x - h)^2 + (y - k)^2 = r^2
-
B.
x^2 + y^2 = r^2
-
C.
x^2 + y^2 + Dx + Ey + F = 0
-
D.
y = mx + b
Solution
The equation x^2 + y^2 + Dx + Ey + F = 0 represents a family of circles.
Correct Answer: C — x^2 + y^2 + Dx + Ey + F = 0
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Q. The general form of the family of curves y^2 = 4ax is known as:
-
A.
Circle
-
B.
Ellipse
-
C.
Parabola
-
D.
Hyperbola
Solution
The equation y^2 = 4ax represents a parabola.
Correct Answer: C — Parabola
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