Question

5. (5 marks) Decide whether the following statements are true or false. (a) If $f(a)f(b) < 0$ and $f(x) \neq 0$ for $x \in (a, b)$, then $f$ is not continuous on $[a, b]$. true false (b) If $\lim_{x \to 1^+} f(x) = 2 = \lim_{x \to 1^-} f(x)$, then $f$ is continuous at $x = 1$. true false (c) If $f$ is differentiable at $x = a$ then $f$ is continuous at $x = a$. true false (d) $\int_2^2 \sqrt{x^3 + 7} \, dx = 0$ true false (e) For all $f, g$: $\int f(x)g(x) \, dx = \left( \int f(x) \, dx \right) \left( \int g(x) \, dx \right)$ true false

          5. (5 marks) Decide whether the following statements are true or false.
(a) If $f(a)f(b) < 0$ and $f(x) \neq 0$ for $x \in (a, b)$, 
then $f$ is not continuous on $[a, b]$.
true
false
(b) If $\lim_{x \to 1^+} f(x) = 2 = \lim_{x \to 1^-} f(x)$, then $f$ is continuous at $x = 1$.
true
false
(c) If $f$ is differentiable at $x = a$ then $f$ is continuous at $x = a$.
true
false
(d) $\int_2^2 \sqrt{x^3 + 7} \, dx = 0$
true
false
(e) For all $f, g$: $\int f(x)g(x) \, dx = \left( \int f(x) \, dx \right) \left( \int g(x) \, dx \right)$
true
false

5. (5 marks) Decide whether the following statements are true or false.
(a) If f(a)f(b) < 0 and f(x) ≠ 0 for x ∈ (a, b),
then f is not continuous on [a, b].
true
false
(b) If limx → 1^+ f(x) = 2 = limx → 1^- f(x), then f is continuous at x = 1.
true
false
(c) If f is differentiable at x = a then f is continuous at x = a.
true
false
(d) ∫2^2 √(x^3 + 7) dx = 0
true
false
(e) For all f, g: ∫ f(x)g(x) dx = ( ∫ f(x) dx ) ( ∫ g(x) dx )
true
false

Added by Christina B.

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