Question: Which of the following is not always true?
a. $\int_c^b f(x)dx = \int_a^b f(x)dx - \int_a^c f(x)dx$
b. If $f(x) \ge g(x)$ for $x \in [a,b]$ then $\int_a^b f(x)dx \ge \int_a^b g(x)dx$
c. $\int_a^b cf(x) + dg(x)dx = c\int_a^b f(x)dx + d\int_a^b g(x)dx$
d. $\int_a^b f(x)dx = -\int_b^a f(x)dx$
e. $|\int_a^b f(x)dx| = \int_a^b |f(x)|dx$
Which of the following statements is true regarding Riemann sums if $f$ is a decreasing function?
$\circ$ a. The midpoint Riemann Sum approximation will be less than the right Riemann Sum approximation.
$\circ$ b. The right Riemann Sum approximation would be less than the left Riemann Sum approximation.
$\circ$ c. Increasing the number of subintervals will decrease the right Riemann Sum.
$\circ$ d. The left Riemann Sum rule would approximate the area better than the midpoint Riemann sum rule.
$\circ$ e. None of the answers
Let $F(x) = \int_x^9 e^{-t} \ln(t^5 - 8t) dt$. Which of the following will give $F'(x)$?
$\circ$ a. $-e^{-x} \ln(x^5 - 8x)$
$\circ$ b. $-e^{-x} \ln(x^5 - 8x) + \frac{e^{-x}(5x^4-8)}{x^5-8x}$
$\circ$ c. None of the answers
$\circ$ d. $e^{-x} \ln(x^5 - 8x)$
$\circ$ e. $e^{-9} \ln(58977) - e^{-x} \ln(x^5 - 8x)$
What is the following statements is true about the Fundamental Theorem of Calculus?
a. The definite integral produces a function while the indefinite integral produces a real number value.
b. None of the answers
c. If $f$ and $g$ are functions satisfying $f'(x) = g'(x)$ for all $x \in [a, b]$, it must follow that $f(x) = g(x)$ for all $x \in [a, b]$.
d. Only functions that are differentiable are Riemann integrable.
e. For any real numbers $a$ and $b$, $\frac{d}{dx} \left[\int_a^b f(x)dx\right] = f(x)$
1. **Problem 1:** Identify which integral property is not always true.
2. **Problem 2:** Determine the true statement about Riemann sums for a decreasing function.
3. **Problem 3:** Find the derivative $F'(x)$ of $F(x) = \int_x^9 e^{-t} \ln(t^5 - 8t) dt$.
4. **Problem 4:** Identify the true statement about the Fundamental Theorem of Calculus.
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### Problem 1
**Step 1:** Recall integral properties:
- Property a: $\int_c^b f(x)dx = \int_a^b f(x)dx - \int_a^c f(x)dx$ is true by splitting integrals.
- Property b: If $f(x) \ge g(x)$ on $[a,b]$, then $\int_a^b f(x)dx \ge \int_a^b g(x)dx$ is true by comparison.
- Property c: Linearity of integrals: $\int_a^b (cf(x) + dg(x)) dx = c\int_a^b f(x)dx + d\int_a^b g(x)dx$ is true.
- Property d: Reversing limits changes sign: $\int_a^b f(x)dx = -\int_b^a f(x)dx$ is true.
- Property e: $|\int_a^b f(x)dx| = \int_a^b |f(x)|dx$ is **not always true** because absolute value of integral is not equal to integral of absolute value.
**Answer:** e is not always true.
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### Problem 2
Given $f$ decreasing on $[a,b]$:
- Left Riemann sum overestimates area.
- Right Riemann sum underestimates area.
- Midpoint sum is more accurate than left or right.
Check options:
- a. Midpoint sum less than right sum? False, midpoint is usually better.
- b. Right sum less than left sum? True for decreasing $f$.
- c. Increasing subintervals decreases right sum? False, right sum approaches true integral.
- d. Left sum better than midpoint? False.
- e. None of the answers? False.
**Answer:** b.
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### Problem 3
Given $F(x) = \int_x^9 e^{-t} \ln(t^5 - 8t) dt$.
By Leibniz rule for differentiation under integral with variable limit:
$$F'(x) = - e^{-x} \ln(x^5 - 8x)$$
Check options:
- a matches this.
- b adds extra term, incorrect.
- c none.
- d positive sign, incorrect.
- e unrelated.
**Answer:** a.
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### Problem 4
Check statements about Fundamental Theorem of Calculus (FTC):
- a. Definite integral produces a number, indefinite integral produces a function (opposite of statement). False.
- b. None of the answers.
- c. If $f'(x) = g'(x)$ on $[a,b]$, then $f(x) = g(x) + C$, not necessarily equal. False.
- d. Only differentiable functions are Riemann integrable? False, integrability requires boundedness and limited discontinuities.
- e. $\frac{d}{dx} \left[\int_a^b f(x) dx\right] = f(x)$ is false because integral limits are constants; derivative is zero.
**Answer:** b.
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### Final answers:
1. Not always true: e
2. True Riemann sum statement: b
3. $F'(x) = - e^{-x} \ln(x^5 - 8x)$ (option a)
4. True FTC statement: b