Derivatives - left and right derivatives, mean value theorem, Rolle's theorem, Taylor's theorem, L'Hôpital's rule - One Line Questions

1. If lim (x->c) f(x) = 0 and lim (x->c) g(x) = 0, and lim (x->c) f'(x)/g'(x) exists, what is lim (x->c) f(x)/g(x)? — lim (x->c) f'(x)/g'(x)
2. Evaluate the limit: lim (x->0) sin(x)/x. — 1
3. Evaluate the limit: lim (x->infinity) x/e^x. — 0
4. Evaluate lim (x->0) (e^x - 1 - x)/x^2. — 1/2
5. Evaluate lim (x->0) (tan(x) - x)/x^3. — 1/3
6. If lim (x->c) f(x)/g(x) is of the form 0/0, and lim (x->c) f'(x)/g'(x) = L, then lim (x->c) f(x)/g(x) = ? — L
7. Which of the following is an indeterminate form for which L'Hôpital's Rule can be applied? — 0/0
8. Which of the following is NOT an indeterminate form for L'Hôpital's Rule? — 1^0
9. Consider the function f(x) = |x|. What is the left-hand derivative of f at x = 0? — -1
10. Consider the function f(x) = |x|. What is the right-hand derivative of f at x = 0? — 1
11. What is the left-hand derivative of f(x) = x^2 at x = 1? — 2
12. What is the right-hand derivative of f(x) = x^2 at x = 1? — 2
13. For f(x) = x^3 on [-1, 1], Rolle's Theorem guarantees a value c in (-1, 1) such that f'(c) = ?. — 0
14. What is the left-hand derivative of f(x) = x|x| at x = 0? — 0
15. What is the right-hand derivative of f(x) = x|x| at x = 0? — 0
16. What is the Taylor expansion of cos(x) around a = 0 up to the second-degree term? — 1 - x^2/2!
17. Which of the following is the Taylor expansion of e^x around a = 0 up to the third-degree term? — 1 + x + x^2/2! + x^3/3!
18. If a function f satisfies the conditions of Rolle's Theorem on [a, b], what is guaranteed to exist? — A point c in (a, b) where f'(c) = 0.
19. The statement of Taylor's Theorem requires the function to have n+1 continuous derivatives on the interval. — True for Lagrange form of remainder
20. Let f(x) = 3x^2 + 2x + 1 on [0, 1]. According to Rolle's Theorem, find c such that f'(c) = 0. — c = -1/3
21. If f(x) = x^3 - x, find c in (-1, 1) such that f'(c) = 0, according to Rolle's Theorem. — c = 1/sqrt(3)
22. If f(x) = x^2 on [1, 3], find c such that f'(c) = (f(3) - f(1)) / (3 - 1). — c = 2
23. Consider f(x) = 1/x on [1, 2]. Find c such that f'(c) = (f(2) - f(1))/(2-1). — c = sqrt(2)
24. Let f(x) = x^3 on the interval [0, 2]. According to the Mean Value Theorem, what is a possible value of c in (0, 2) such that f'(c) = (f(2) - f(0)) / (2 - 0)? — c = 2 / sqrt(3)
25. Consider f(x) = ln(x) on [1, e]. Find c in (1, e) such that f'(c) = (f(e) - f(1))/(e-1). — c = 1/ (e-1)
26. Consider the limit lim (x->0) (1 - cos(x))/x^2. What is the first step in applying L'Hôpital's Rule? — Check if the form is indeterminate.
27. If a function f is analytic at a point a, what does Taylor's Theorem imply about its representation near a? — f can be represented by its Taylor series, which converges to f.
28. If the left-hand derivative and right-hand derivative of f at c exist and are equal, what can be concluded about f at c? — f is differentiable at c.
29. Which condition is NOT required for Rolle's Theorem to be applicable to a function f on the interval [a, b]? — f'(x) exists for all x in [a, b].
30. If f'(x) = 0 for all x in an interval (a, b), what can be said about the function f on that interval? — f is a constant function.
31. In Taylor's theorem f(x) = P_n(x) + R_n(x), where P_n(x) is the Taylor polynomial and R_n(x) is the remainder term. What is the Lagrange form of the remainder R_n(x) for a function f with (n+1) derivatives? — f^(n+1)(c)(x-a)^(n+1) / (n+1)! for some c between a and x.
32. Rolle's Theorem can be considered a special case of the Mean Value Theorem. What is the specific condition that makes MVT reduce to Rolle's Theorem? — f(a) = f(b)
33. What is the statement of Taylor's Theorem with the Lagrange form of the remainder? — f(x) = f(a) + f'(a)(x-a) + ... + f^(n-1)(a)(x-a)^(n-1) / (n-1)! + f^(n)(c)(x-a)^n / n!
34. What is L'Hôpital's Rule primarily used for? — Evaluating indeterminate forms of limits.
35. What is the statement of the Mean Value Theorem (MVT)? — If f is continuous on [a, b] and differentiable on (a, b), then there exists at least one c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).
36. What is the statement of Rolle's Theorem? — If f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists at least one c in (a, b) such that f'(c) = 0.
37. Which of the following indeterminate forms can be transformed into 0/0 or infinity/infinity to apply L'Hôpital's Rule? — All of the above
38. If lim (x->c) f'(x)/g'(x) does not exist, what can be concluded about lim (x->c) f(x)/g(x)? — It cannot be determined by L'Hôpital's Rule alone.
39. Let f(x) = x^2 - 4x + 3 on the interval [1, 3]. Which theorem can be applied here, and what is a possible value of c? — Rolle's Theorem; c = 2
40. What is the Maclaurin series, which is a special case of Taylor series? — Taylor series expansion of a function around a = 0.
41. For a function f to be differentiable at a point c, which of the following conditions must be met? — The left-hand derivative must equal the right-hand derivative.
42. State the condition for applying L'Hôpital's Rule to the limit of f(x)/g(x) as x approaches c. — The limit must be of the form 0/0 or infinity/infinity, and g'(x) is not zero near c.
43. What is the definition of the left-hand derivative of a function f at a point c? — The limit of (f(c) - f(c-h)) / h as h approaches 0 from the left.
44. What is the definition of the right-hand derivative of a function f at a point c? — The limit of (f(c+h) - f(c)) / h as h approaches 0 from the right.
45. Taylor's Theorem is particularly useful for approximating the value of a function near a specific point. What is the primary use of the remainder term? — To bound the error in the approximation.
46. The Mean Value Theorem implies that the slope of the tangent line at some point c is equal to the slope of the secant line connecting the endpoints of the interval. — True
47. The remainder term in Taylor's theorem provides an upper bound for the error when approximating f(x) by its Taylor polynomial. — True
48. What is the Taylor expansion of sin(x) around a = 0 up to the third-degree term? — x - x^3/3!
49. Let f(x) = sqrt(x) on [0, 1]. Can Rolle's Theorem be applied? — No, because f is not differentiable at x=0.
50. If f(x) = x^3 - 6x^2 + 5 on [0, 4], does Rolle's Theorem apply? If so, find c. — No, because f(0) != f(4)
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