Q: Suppose $$g : (0, 1) \to \mathbb{R} is $$continuous and $$g(x) = \frac{1}{x}. $$Why does $$g$$ not attain a maximum on $$(0, 1)$$?

Because as $$x \to 0^+$$, $$g(x) \to \infty$$, so there is no maximum value.

Question 1

Given a function $$h : [0, 1] 	o \mathbb{R}$$, what is the Riemann sum for $$h on$$ $$[0, 1]$$ divided into four equal-width sub-intervals, using left endpoints as sample points? Express your answer as a sum of four terms.

Accepted Answer

$$h(0) \cdot \frac{1}{4} + h(\frac{1}{4}) \cdot \frac{1}{4} + h(\frac{1}{2}) \cdot \frac{1}{4} + h(\frac{3}{4}) \cdot \frac{1}{4}$$

Question 2

Let $$f(t) = t - 5$$ for $$t \in [0, 5]. $$What is the value of $$\$$int_0$$^5 f(t) \, dt$$?

Accepted Answer

$$-12.5$$

Question 3

Which of the following is the correct statement of the Extreme Value Theorem?

Accepted Answer

If $$f is $$continuous on a closed, bounded interval $$[a, b]$$, then $$f$$ attains both a maximum and a minimum value on $$[a, b].$$

Question 4

Suppose $$g : (0, 1) 	o \mathbb{R} is $$continuous and $$g(x) = \frac{1}{x}. $$Why does $$g$$ not attain a maximum on $$(0, 1)$$?

Accepted Answer

Because as $$x 	o 0^+$$, $$g(x) 	o \infty$$, so there is no maximum value.

Question 5

Let $$p, q : \mathbb{N} 	o \mathbb{R}$$ with $$p(i) \leq q(i)$$ for all $$i \in \mathbb{N}. $$What does induction show about the sums $$\sum_{i=1}^n p(i)$$ and $$\sum_{i=1}^n q(i)$$?

Accepted Answer

$$\sum_{i=1}^n p(i) \leq \sum_{i=1}^n q(i)$$ for all $$n \in \mathbb{N}.$$

Question 6

Given $$f : [a, b] 	o \mathbb{R}$$ continuous, and a partition $$a = x_0$ \u003c x_1$ \u003c \dots \u003c x_n = b$$, with $$\Delta x_i = x_i - x_{i-1}$$, $$f(m_i) = \min\{f(t) : t \in [x_{i-1}, x_i]\}$$, $$f(M_i) = \max\{f(t) : t \in [x_{i-1}, x_i]\}$$, and $$x_i^* \in [x_{i-1}, x_i]$$, which inequality holds for all $$i$$?

Accepted Answer

$$f(m_i) \leq f(x_i^*) \leq f(M_i)$$

Question 7

A rocket nozzle is formed by revolving the area between $$y = \sqrt{x}$$ and $$y = x$$ for $$x \in [0, 1]$$ about the $$x-$$axis. What is the volume of material required?

Accepted Answer

$$\$$int_0$$^1 \pi \left((\sqrt{x})^2$ - (x)^2$\right) dx$$

Question 8

A car travels 100 km in 0.5 hours. Using the Mean Value Theorem for integrals, what can you conclude about the car's instantaneous speed during the trip?

Accepted Answer

There exists a time when the instantaneous speed was exactly 200 km/h.

Question 9

Let $$G(x) = e$$^{-\int_0^x f(t) dt}$$ for a continuous function f$ on $$[0, T]$$. What is G$$'(x)$$?

Accepted Answer

$$-f(x) G(x)$$

Question 10

Consider the solid sphere of radius $$R$$ centered at the origin. What is the radius of the disc formed by slicing the sphere with the plane $$x = t$$?

Accepted Answer

$$\sqrt{R^2$ - t^2$}$$

Question 11

What is the area of the disc formed by the intersection of the sphere of radius $$R$$ and the plane $$x = t$$?

Accepted Answer

$$\pi (R^2$ - t^2$)$$

Question 12

What is the volume of the sphere of radius $$R$$ computed by integrating the cross-sectional area from $$t = -R to$$ $$t = R$$?

Accepted Answer

$$\int_{-R}^R \pi (R^2$ - t^2$) dt = \frac{4}{3} \pi R^3$

Question 13

Let $$F(x) = y_0$ + \$$int_0$$^x f(t) dt$$ for a continuous function $$f on$$ $$[0, T]. $$Which initial value problem does $$F$$ solve?

Accepted Answer

$$y'(x) = f(x), \quad y(0) = y_0$

Question 14

Let $$y = \sqrt{\cos x}$$ and $$y = 1$$ bound a region for $$x \in [-\frac{\pi}{2}, \frac{\pi}{2}]. $$What is the volume of revolution about the $$x-$$axis?

Accepted Answer

$$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \pi (1^2$ - (\sqrt{\cos x})^2$) dx$$

Calculus Homework Guidance: Riemann Sums, Integrals, Theorems, and Volumes

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