MTH019 Final Review Exercise 20-21
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MTH019 Final Review Exercise 20-21
Questions
I. Multiple Choice Questions.
x2 + 3x
(A) 1. (B) u3. (C) . (D) 0.
北 → _3 x2 u x u 12
2. If the function f (x) =., 0北 (et x(u) 4) dt , x 0 is continuous on (u-, -), then a = [ ]
(A) 3. (B) u3. (C) 1. (D) u1.
3. If f is differentiable at x = c, then which of the following is NOT true? [ ]
f (x) u f (c) f (c + h) u f (c)
北 →c x u c h →0 h
(C) lim = f/ (c). (D) lim = f/ (c).
4. If f (x) = tan x , then f/ ( 4 ) = [ ]
(A) 2. (B) 2 . (C) 1 u 2 . (D) 1 + 2 .
1 π π
5. Let the point P (u^3, y0 ) be on the graph of y = arctan x. Find the equation of the tangent line to
the graph of y = arctan x at P . [ ]
(A) y u π = (x +^3). (B) y u π = u (x +^3).
(C) y + π = (x +^3). (D) y + π = u (x +^3).
6. If y = x3 , then find the value of dy as x changes from 2 to 2.05. [ ]
(A) 0.000125. (B) 0.62. (C) 0.60. (D) 0.61.
7. The minimum value of the function f (x) = x2 +北(16) on [1, 4]is [ ]
(A) 17. (B) 12. (C) 20. (D) 8.
8. If the graph of y = x3 + ax2 + bx u 4 has the inflection point (1, u6), what is the value of b? [ ] (A) u3. (B) 3. (C) 1. (D) 0.
9. Let f be a differentiable function and f// (x) < 0 for all real numbers x. Then [ ]
(A) f/ (1) > f/ (0) > f (1) u f (0). (B) f/ (0) > f (1) u f (0) > f/ (1).
(C) f/ (1) > f (1) u f (0) > f/ (0). (D) f/ (0) > f/ (1) > f (1) u f (0).
尸
10. lim (1 + 5e北 ) 达 = [ ] 北 →2
(A) 0. (B) 1. (C) e. (D) e5 .
11. The velocity of an object is v = 9.8t + 5 which moves along a coordinate line. Its initial position is given by s(0) = 10. Then the position at time t is [ ] .
(A) s = 5t. (B) s = 10. (C) s = 4.9t2 + 5t + 10. (D) s = 4.9t2 + 5t.
5 8 8 8
12. Suppose that f (x) dx = 3, f (x) dx = 4 and g(x) dx = 7. Then ╱3f (x) + 2g(x)、dx = 2 5 2 2
[ ]
(A) 23. (B) 26. (C) 35. (D) 42.
5
_5
(A) 2(e5 u 1). (B) e5 u e_5 . (C) 0. (D) e5 u 1.
π/2 cos θ
0 ^1 + sin θ
(A) 2. (B) 2^2. (C) 2(^2 u 1). (D) 2(^2 + 1).
2
北
15. If F (x) = ^1 + t3 dt, then F\ (x) = [ ] 1
(A) ^1 + x3 . (B) ^1 + x6 . (C) 2x^1 + x3 . (D) 2x^1 + x6 .
3^3 x3
0 ^9 + x2
(A) 27. (B) 45. (C) 18. (D) 36.
17. Find the length of the curve y = x + 5 from x = 0 to x = 8. [ ] (A) . (B) . (C) + 40. (D) 40.
18. Which of the following statements is TRUE? [ ]
(A) The function f (x) = ^x satisfies the hypotheses of the Mean Value Theorem on [0 , 2]. (B) If f\ (c) = f\\ (c) = 0, then f (c) is neither a maximum nor minimum value.
(C) The product of two increasing functions is an increasing function.
(D) Suppose that f (0) = 5 and that f\ (x) = 2 for all x. Then f (x) = 2x for all x.
1
19. xe北 dx = [ ]
0
(A) e. (B) 2e u 1. (C) 1. (D) 0.
20. The particular solution of x u y = x3 that satisfies y = 1 when x = 1 is [ ] (A) y = x3 + x. (B) y = x4 + x.
(C) y = u x4 ln x + x3 + . (D) y = x4 ln x u x3 + .
II. Comprehensive problems.
21. Let f (x) = x +^x2 + 2x.
(1) Find the natural domain of f .
(2) Determine where f is increasing and where f is decreasing.
(3) Find the range of f .
22. As shown in the Figure 1, the curve C is the graph of the function f (x) = 3/x and L is the tangent line to the curve C at x = a. The tangent line intersects the x-axis at the point A and y-axis at the point B .
(1) Find the equation for the tangent line L.
(2) Find the shortest length of the line segment AB .
Figure 1
23. Let R be the region in the first quadrant enclosed by the graphs of f (x) = 8x3 and g(x) = sin(πx), as shown in the Figure 2.
(1) Find the area of R.
(2) Find the volume of the solid of revolution generated by revolving R about x-axis.
(3) Find the volume of the solid of revolution generated by revolving R about y-axis.
Figure 2
24. Three curves C1 : y = 2x2 , C2 : y = x2 and C lie in the first quadrant as shown in the Figure 3. Let P (a, 2a2 ), ( a > 0) be an arbitrary point on the curve C1 . The point P1 is on C2 such that the line PP1 is parallel to the y-axis, and the point P2 is on C such that the line PP2 is parallel to the x-axis. Let A denote the region bounded by C1 , C2 and PP1 , and let B denote the region bounded by C, C1 and PP2 . Let SA and SB be the area of the region A and the region B respectively. If SA = SB for all a > 0, find the equation of the curve C .
Figure 3
2022-12-15