Q. For the function f(x) = x^2 - 2x + 1, find the slope of the tangent line at x = 1.
Solution
f'(x) = 2x - 2. Thus, f'(1) = 2(1) - 2 = 0.
Correct Answer: A — 0
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Q. For the function f(x) = x^2 - 4x + 4, find the point where it is not differentiable.
-
A.
x = 0
-
B.
x = 2
-
C.
x = 4
-
D.
It is differentiable everywhere
Solution
As a polynomial, f(x) is differentiable everywhere, including at x = 2.
Correct Answer: D — It is differentiable everywhere
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Q. For the function f(x) = x^3 - 3x^2 + 2, find the points where it is not differentiable.
-
A.
None
-
B.
x = 0
-
C.
x = 1
-
D.
x = 2
Solution
As a polynomial, f(x) is differentiable everywhere, hence no points of non-differentiability.
Correct Answer: A — None
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Q. For the function f(x) = x^3 - 3x^2 + 4, find the points where it is not differentiable.
-
A.
None
-
B.
x = 0
-
C.
x = 1
-
D.
x = 2
Solution
The function is a polynomial and is differentiable everywhere, hence there are no points where it is not differentiable.
Correct Answer: A — None
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Q. For the function f(x) = x^3 - 3x^2 + 4, find the value of x where f is not differentiable.
Solution
The function is a polynomial and is differentiable everywhere, so there is no such x.
Correct Answer: A — 0
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Q. For the function f(x) = x^3 - 3x^2 + 4, find the x-coordinate of the point where f is differentiable.
Solution
f(x) is a polynomial and is differentiable everywhere. The x-coordinate can be any real number.
Correct Answer: C — 1
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Q. For the function f(x) = { x^2, x < 1; kx + 1, x >= 1 }, find k such that f is differentiable at x = 1.
Solution
Setting f(1-) = f(1+) and f'(1-) = f'(1+) gives k = 2 for differentiability.
Correct Answer: B — 1
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Q. For the function f(x) = |x - 2| + |x + 3|, find the point where it is not differentiable.
Solution
The function is not differentiable at x = -3 and x = 2, but the first point of interest is -3.
Correct Answer: A — -3
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Q. For which value of a is the function f(x) = x^2 + ax + 1 differentiable at x = -1?
Solution
To ensure differentiability at x = -1, we find f'(-1) exists. Setting a = 0 ensures the derivative is defined.
Correct Answer: B — 0
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Q. For which value of a is the function f(x) = x^2 + ax + 1 differentiable everywhere?
Solution
The function is a polynomial and is differentiable for all real numbers, hence any value of a works.
Correct Answer: B — 0
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Q. For which value of a is the function f(x) = x^2 - ax + 2 differentiable at x = 1?
Solution
Setting the derivative f'(1) = 0 gives a = 1 for differentiability.
Correct Answer: B — 1
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Q. For which value of a is the function f(x) = x^2 - ax + 4 differentiable at x = 2?
Solution
f(x) is a polynomial and is differentiable for all a, hence any value of a works.
Correct Answer: A — 0
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Q. For which value of a is the function f(x) = x^3 - 3ax + 2 differentiable at x = 1?
Solution
Setting f'(1) = 0 gives a = 1, ensuring differentiability at that point.
Correct Answer: B — 1
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Q. For which value of a is the function f(x) = x^3 - 3ax^2 + 3a^2x + 1 differentiable at x = 1?
Solution
Setting f'(1) = 0 gives a = 1 for differentiability at x = 1.
Correct Answer: B — 1
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Q. For which value of k is the function f(x) = kx^2 + 2x + 1 differentiable at x = -1?
Solution
f'(x) = 2kx + 2; f'(-1) = -2k + 2 must exist for any k.
Correct Answer: A — 0
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Q. If f(x) = e^x, then f'(0) is equal to?
Solution
f'(x) = e^x; f'(0) = e^0 = 1.
Correct Answer: B — 1
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Q. If f(x) = ln(x) for x > 0, is f differentiable at x = 1?
-
A.
Yes
-
B.
No
-
C.
Only continuous
-
D.
Only left differentiable
Solution
f'(x) = 1/x; f'(1) = 1, hence f is differentiable at x = 1.
Correct Answer: A — Yes
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Q. If f(x) = x^2 + 2x + 1 for x < 0 and f(x) = kx + 1 for x >= 0, find k such that f is differentiable at x = 0.
Solution
Setting the left-hand derivative equal to the right-hand derivative at x = 0 gives k = 2.
Correct Answer: A — -1
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Q. If f(x) = x^2 + 2x + 1, find f'(1).
Solution
f'(x) = 2x + 2, thus f'(1) = 2(1) + 2 = 4.
Correct Answer: C — 3
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Q. If f(x) = x^2 + 2x + 1, what is f'(1)?
Solution
Calculating the derivative f'(x) = 2x + 2, we find f'(1) = 4.
Correct Answer: B — 3
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Q. If f(x) = x^2 + 2x + 3, find f'(1).
Solution
f'(x) = 2x + 2. Therefore, f'(1) = 2(1) + 2 = 4.
Correct Answer: C — 4
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Q. If f(x) = x^2 - 4x + 4, find f'(2).
Solution
f'(x) = 2x - 4. Thus, f'(2) = 2(2) - 4 = 0.
Correct Answer: A — 0
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Q. If f(x) = x^2 for x < 1 and f(x) = 2x - 1 for x ≥ 1, is f differentiable at x = 1?
-
A.
Yes
-
B.
No
-
C.
Only continuous
-
D.
Only left differentiable
Solution
f'(1) from left = 2 and from right = 2; hence f is differentiable at x = 1.
Correct Answer: B — No
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Q. If f(x) = x^2 sin(1/x) for x ≠ 0 and f(0) = 0, is f differentiable at x = 0?
-
A.
Yes
-
B.
No
-
C.
Only left differentiable
-
D.
Only right differentiable
Solution
Using the limit definition of the derivative, f'(0) exists, hence f is differentiable at x = 0.
Correct Answer: A — Yes
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Q. If f(x) = x^3 - 3x + 2, find the critical points where f'(x) = 0.
Solution
Set f'(x) = 3x^2 - 3 = 0 and solve for x.
Correct Answer: B — 0
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Q. If f(x) = x^3 - 3x + 2, find the points where f is not differentiable.
Solution
The function is a polynomial and is differentiable everywhere, hence no points of non-differentiability.
Correct Answer: A — 0
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Q. If f(x) = x^3 - 3x^2 + 4, find the point where f is not differentiable.
Solution
The function is a polynomial and is differentiable everywhere, but checking critical points shows f'(x) = 0 at x = 2.
Correct Answer: C — 2
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Q. If f(x) = x^3 - 3x^2 + 4, then f'(1) is equal to?
Solution
f'(x) = 3x^2 - 6x; f'(1) = 3(1)^2 - 6(1) = 3 - 6 = -3.
Correct Answer: B — 2
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Q. If f(x) = x^3 - 3x^2 + 4, then f'(2) is equal to?
Solution
f'(x) = 3x^2 - 6x; f'(2) = 3(2^2) - 6(2) = 12 - 12 = 0.
Correct Answer: B — 1
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Q. If f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1, find f'(1).
Solution
Calculating f'(x) = 4x^3 - 12x^2 + 12x - 4. Thus, f'(1) = 4 - 12 + 12 - 4 = 0.
Correct Answer: A — 0
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