Polynomial rational trigonometric logarithmic and exponential functions - Question Bank

1. The function f(x) = 1/x is:
A) Continuous everywhere.
B) Differentiable everywhere.
C) Continuous for all x != 0, but not differentiable at x = 0.
D) Discontinuous at x = 0.
2. What is the derivative of f(x) = sqrt(x^2 + 1)?
A) x / sqrt(x^2 + 1)
B) 1 / (2*sqrt(x^2 + 1))
C) sqrt(2x)
D) x
3. What is the limit of (x - 5) / (x^2 - 25) as x approaches 5?
A) 0
B) 1/10
C) 1/5
D) Undefined
4. If a function is differentiable at a point, it must be:
A) Constant
B) Zero
C) Continuous
D) Increasing
5. Find the derivative of f(x) = cos(3x).
A) -sin(3x)
B) -3sin(3x)
C) 3cos(3x)
D) sin(3x)
6. What is the limit of (3x^2 + 2x - 1) / (x^2 - 5) as x approaches infinity?
A) 0
B) 1
C) 3
D) Undefined
7. Consider the function f(x) = x^2 sin(1/x) for x != 0 and f(0) = 0. Is f(x) differentiable at x = 0?
A) Yes, the derivative is 0.
B) No, because sin(1/x) is not defined at x=0.
C) Yes, because the limit of the difference quotient exists.
D) No, because the function is not continuous at x=0.
8. What is the derivative of f(x) = x^x?
A) x^x * (1 + ln(x))
B) x * x^(x-1)
C) x^x
D) x^x * ln(x)
9. Evaluate the limit of (ln(x)) as x approaches 1.
A) 0
B) 1
C) e
D) Undefined
10. The function f(x) = x^3 - x is:
A) Continuous only at x = 0.
B) Differentiable only at x = 0.
C) Continuous and differentiable for all real numbers.
D) Discontinuous at x = 1 and x = -1.
11. Find the derivative of f(x) = e^(sin(x)).
A) e^(sin(x))
B) cos(x) * e^(sin(x))
C) sin(x) * e^(cos(x))
D) e^(cos(x))
12. What is the limit of (x^2) as x approaches infinity?
A) 0
B) 1
C) infinity
D) -infinity
13. Which of the following is NOT a property of limits?
A) lim (x->c) [f(x) + g(x)] = lim (x->c) f(x) + lim (x->c) g(x)
B) lim (x->c) [f(x) * g(x)] = lim (x->c) f(x) * lim (x->c) g(x)
C) lim (x->c) [f(x) / g(x)] = [lim (x->c) f(x)] / [lim (x->c) g(x)] if lim (x->c) g(x) != 0
D) lim (x->c) f(g(x)) = f(lim (x->c) g(x)) if f is continuous at lim (x->c) g(x)
14. If f(x) = x * |x|, what is f'(0)?
A) 1
B) -1
C) 0
D) Undefined
15. What is the derivative of f(x) = log_a(x^2)?
A) 2/(x*ln(a))
B) 1/(x*ln(a))
C) 2x/ln(a)
D) 2/(x*log_a(e))
16. Evaluate the limit of (sin(ax)) / (bx) as x approaches 0, where a and b are non-zero constants.
A) 0
B) 1
C) a/b
D) b/a
17. The function f(x) = x^3 is:
A) Continuous everywhere but not differentiable at x = 0.
B) Continuous and differentiable everywhere.
C) Discontinuous at x = 0.
D) Differentiable only for x > 0.
18. Find the derivative of f(x) = arctan(x).
A) 1/(1 + x^2)
B) -1/(1 + x^2)
C) 1/sqrt(1 - x^2)
D) 1/x
19. What is the limit of (x^n - a^n) / (x - a) as x approaches a?
A) a^n
B) na^(n-1)
C) n*a^n
D) Undefined
20. If lim (x->a) f(x) = L and f(a) = L, then f(x) is:
A) Differentiable at x = a.
B) Continuous at x = a.
C) Undefined at x = a.
D) Discontinuous at x = a.
21. Find the derivative of f(x) = sin(x^2).
A) cos(x^2)
B) 2x*cos(x^2)
C) cos(2x)
D) 2x*sin(x^2)
22. What is the limit of (e^(2x) - 1) / x as x approaches 0?
A) 0
B) 1
C) 2
D) Undefined
23. The function f(x) = |x - 2| is:
A) Continuous but not differentiable at x = 2.
B) Differentiable at x = 2.
C) Neither continuous nor differentiable at x = 2.
D) Continuous and differentiable at x = 2.
24. What is the derivative of arccos(x)?
A) -1/sqrt(1 - x^2)
B) 1/sqrt(1 - x^2)
C) 1/sqrt(x^2 - 1)
D) -1/(1 + x^2)
25. Evaluate the limit of tan(x) / x as x approaches 0.
A) 0
B) 1
C) infinity
D) -1
26. A function f(x) is differentiable at x = c if:
A) lim (h->0) [f(c+h) - f(c)] / h exists.
B) f(x) is continuous at x = c.
C) f'(c) exists.
D) All of the above.
27. Find the derivative of f(x) = (x^2 + 1) / (x - 1) using the quotient rule.
A) (x^2 - 2x - 1) / (x - 1)^2
B) (x^2 + 2x + 1) / (x - 1)^2
C) (2x) / 1
D) (x^2 - 2x + 1) / (x - 1)^2
28. What is the limit of (sqrt(x) - 2) / (x - 4) as x approaches 4?
A) 0
B) 1/4
C) 1/2
D) Undefined
29. Consider the function f(x) = x^2 if x is rational and f(x) = 0 if x is irrational. At which point is f(x) continuous?
A) x = 0
B) x = 1
C) All points
D) No points
30. If f(x) = a^x, where a > 0 and a != 1, what is its derivative?
A) x*a^(x-1)
B) a^x
C) a^x * ln(a)
D) a^x / ln(a)
31. What is the derivative of arcsin(x)?
A) 1/sqrt(1 - x^2)
B) -1/sqrt(1 - x^2)
C) 1/sqrt(x^2 - 1)
D) 1/(1 + x^2)
32. Evaluate the limit of (1 - cos(x)) / x^2 as x approaches 0.
A) 0
B) 1/2
C) 1
D) Undefined
33. Which condition is NOT required for a function f(x) to be differentiable at x = c?
A) f(x) must be continuous at x = c.
B) The limit of the difference quotient must exist.
C) The derivative from the left must equal the derivative from the right.
D) f(c) must be a real number.
34. Find the second derivative of f(x) = x^3.
A) 3x^2
B) 6x
C) 3
D) 0
35. What is the limit of (x^2 + 5x + 6) / (x + 2) as x approaches -2?
A) -2
B) -1
C) 0
D) Undefined
36. Consider the function f(x) = x^2 for x < 0 and f(x) = x for x >= 0. Is f(x) continuous at x = 0?
A) Yes, because both parts approach 0.
B) No, because the derivative is different from the left and right.
C) Yes, because f(0) = 0 and the limits from both sides are 0.
D) No, because the function definition changes at x = 0.
37. If f(x) = log_b(x), what is its derivative?
A) 1/x
B) 1/(x*ln(b))
C) ln(b)/x
D) x*ln(b)
38. Evaluate the limit of (e^x - 1) / x as x approaches 0.
A) 0
B) 1
C) e
D) Undefined
39. What is the derivative of tan(x)?
A) sec^2(x)
B) -csc^2(x)
C) sec(x)tan(x)
D) -sec(x)tan(x)
40. Which of the following functions is NOT continuous at x = 0?
A) f(x) = x^2
B) f(x) = sin(x)
C) f(x) = 1/x
D) f(x) = e^x
41. Find the derivative of f(x) = 5x^4 - 3x^2 + 2x - 7.
A) 20x^3 - 6x + 2
B) 5x^3 - 3x + 2
C) 20x^4 - 6x^2 + 2x
D) 20x^3 - 6x^2 + 2
42. What is the limit of (x^3 - 1) / (x - 1) as x approaches 1?
A) 0
B) 1
C) 3
D) Undefined
43. Determine if the function f(x) = x^2 is continuous at x = 3.
A) Yes, because f(3) = 9 and lim (x->3) f(x) = 9.
B) No, because the function is a polynomial.
C) Yes, because the function is differentiable.
D) No, because the limit does not exist.
44. If f(x) = e^x, what is the derivative of f(x)?
A) e^x
B) x*e^(x-1)
C) e
D) 1
45. What is the derivative of ln(x) with respect to x?
A) x
B) 1/x
C) ln(x)
D) -1/x
46. Evaluate the limit of sin(x)/x as x approaches 0.
A) 0
B) 1
C) infinity
D) -1
47. What is the derivative of the function f(x) = x^n?
A) nx^(n-1)
B) nx^n
C) x^(n-1)
D) (n-1)x^n
48. If f(x) = |x|, what is the derivative of f(x) at x = 0?
A) 1
B) -1
C) 0
D) Undefined
49. For a function f(x) to be continuous at x = c, which of the following conditions must be met?
A) f(c) must be defined.
B) The limit of f(x) as x approaches c must exist.
C) The limit of f(x) as x approaches c must equal f(c).
D) All of the above.
50. What is the limit of the function f(x) = (x^2 - 4) / (x - 2) as x approaches 2?
A) 0
B) 2
C) 4
D) Undefined
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