Question

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 1. If f and g are continuous on [a, b], then ?_a^b [f(x) + g(x)] dx = ?_a^b f(x) dx + ?_a^b g(x) dx 2. If f and g are continuous on [a, b], then ?_a^b [f(x)g(x)] dx = (?_a^b f(x) dx)(?_a^b g(x) dx) 3. If f is continuous on [a, b], then ?_a^b 5f(x) dx = 5 ?_a^b f(x) dx 4. If f is continuous on [a, b], then ?_a^b xf(x) dx = x ?_a^b f(x) dx 5. If f is continuous on [a, b] and f(x) ? 0, then ?_a^b ?f(x) dx = ??_a^b f(x) dx 6. If f' is continuous on [1, 3], then ?_1^3 f'(v) dv = f(3) ? f(1). 7. If f and g are continuous and f(x) ? g(x) for a ? x ? b, then ?_a^b f(x) dx ? ?_a^b g(x) dx 8. If f and g are differentiable and f(x) ? g(x) for a < x < b, then f'(x) ? g'(x) for a < x < b. 9. ?_{-1}^1 (x^5 - 6x^9 + sin x / (1 + x^4)^2) dx = 0 10. ?_{-5}^5 (ax^2 + bx + c) dx = 2 ?_0^5 (ax^2 + c) dx 11. All continuous functions have derivatives. 12. All continuous functions have antiderivatives. 13. ?_0^3 e^{x^2} dx = ?_0^5 e^{x^2} dx + ?_5^3 e^{x^2} dx 14. If ?_0^1 f(x) dx = 0, then f(x) = 0 for 0 ? x ? 1. 15. If f is continuous on [a, b], then d/dx (?_a^x f(t) dt) = f(x) 16. ?_0^2 (x - x^3) dx represents the area under the curve y = x - x^3 from 0 to 2. 17. ?_{-2}^1 1/x^4 dx = -3/8 18. If f has a discontinuity at 0, then ?_{-1}^1 f(x) dx does not exist.

          Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.
1. If f and g are continuous on [a, b], then ?_a^b [f(x) + g(x)] dx = ?_a^b f(x) dx + ?_a^b g(x) dx
2. If f and g are continuous on [a, b], then ?_a^b [f(x)g(x)] dx = (?_a^b f(x) dx)(?_a^b g(x) dx)
3. If f is continuous on [a, b], then ?_a^b 5f(x) dx = 5 ?_a^b f(x) dx
4. If f is continuous on [a, b], then ?_a^b xf(x) dx = x ?_a^b f(x) dx
5. If f is continuous on [a, b] and f(x) ? 0, then ?_a^b ?f(x) dx = ??_a^b f(x) dx
6. If f' is continuous on [1, 3], then ?_1^3 f'(v) dv = f(3) ? f(1).
7. If f and g are continuous and f(x) ? g(x) for a ? x ? b, then ?_a^b f(x) dx ? ?_a^b g(x) dx
8. If f and g are differentiable and f(x) ? g(x) for a < x < b, then f'(x) ? g'(x) for a < x < b.
9. ?_{-1}^1 (x^5 - 6x^9 + sin x / (1 + x^4)^2) dx = 0
10. ?_{-5}^5 (ax^2 + bx + c) dx = 2 ?_0^5 (ax^2 + c) dx
11. All continuous functions have derivatives.
12. All continuous functions have antiderivatives.
13. ?_0^3 e^{x^2} dx = ?_0^5 e^{x^2} dx + ?_5^3 e^{x^2} dx
14. If ?_0^1 f(x) dx = 0, then f(x) = 0 for 0 ? x ? 1.
15. If f is continuous on [a, b], then d/dx (?_a^x f(t) dt) = f(x)
16. ?_0^2 (x - x^3) dx represents the area under the curve y = x - x^3 from 0 to 2.
17. ?_{-2}^1 1/x^4 dx = -3/8
18. If f has a discontinuity at 0, then ?_{-1}^1 f(x) dx does not exist.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.
1. If f and g are continuous on [a, b], then ?a^b [f(x) + g(x)] dx = ?a^b f(x) dx + ?a^b g(x) dx
2. If f and g are continuous on [a, b], then ?a^b [f(x)g(x)] dx = (?a^b f(x) dx)(?a^b g(x) dx)
3. If f is continuous on [a, b], then ?a^b 5f(x) dx = 5 ?a^b f(x) dx
4. If f is continuous on [a, b], then ?a^b xf(x) dx = x ?a^b f(x) dx
5. If f is continuous on [a, b] and f(x) ? 0, then ?a^b ?f(x) dx = ??a^b f(x) dx
6. If f' is continuous on [1, 3], then ?1^3 f'(v) dv = f(3) ? f(1).
7. If f and g are continuous and f(x) ? g(x) for a ? x ? b, then ?a^b f(x) dx ? ?a^b g(x) dx
8. If f and g are differentiable and f(x) ? g(x) for a < x < b, then f'(x) ? g'(x) for a < x < b.
9. ?-1^1 (x^5 - 6x^9 + sin x / (1 + x^4)^2) dx = 0
10. ?-5^5 (ax^2 + bx + c) dx = 2 ?0^5 (ax^2 + c) dx
11. All continuous functions have derivatives.
12. All continuous functions have antiderivatives.
13. ?0^3 e^x^2 dx = ?0^5 e^x^2 dx + ?5^3 e^x^2 dx
14. If ?0^1 f(x) dx = 0, then f(x) = 0 for 0 ? x ? 1.
15. If f is continuous on [a, b], then d/dx (?a^x f(t) dt) = f(x)
16. ?0^2 (x - x^3) dx represents the area under the curve y = x - x^3 from 0 to 2.
17. ?-2^1 1/x^4 dx = -3/8
18. If f has a discontinuity at 0, then ?-1^1 f(x) dx does not exist.

Added by Deborah C.

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