Suppose that f is a continuous additive function on

$$\mathbb{R}.$$

If c := f (1), show that we have f(x) = cx for all

$$x \in \mathbb{R}.$$

A function

$$f: \mathbb{R}\rightarrow\mathbb{R}$$

is said to be periodic on

$$\mathbb{R}$$

if there exists a number p > 0 such that f(x + p) = f(x) for all

$$x \in\mathbb{R}.$$

Prove that a continuous periodic function on

$$\mathbb{R}$$

is bounded and uniformly continuous on

$$\mathbb{R}.$$

Suppose that a student has 500 vocabulary words to learn. If the student learns 15 words after 5 minutes, the function $L(t) = 500(1 - e^{-0.0061t})$ approximates the number of words L that the student will learn after t minutes. How many words will the student learn after 60 minutes?

Express the integrand as a sum of partial fractions and evaluate the integrals.

$$\int _ { - 1 } ^ { 0 } \frac { x ^ { 3 } d x } { x ^ { 2 } - 2 x + 1 }$$

Compute the first-order partial derivatives.

$$w = \frac { x } { y + z }$$

How many ways can you arrange two of the first three letters of the alphabet?

three siblings are three different ages. The oldest is twice the age of the middle sibling, and the middle sibling is six years older than one-half the age of the youngest. Write a composite function that gives the oldest sibling’s age in terms of the youngest. Explain how you arrived at your answer.

Simplify each sum or difference.

$$( 9 x - 3 y ) - ( 3 x - 9 y )$$

Factor each polynomial.

$$2x^2+7x-30$$

Write each Roman numeral as a Hindu-Arabic numeral.

$$\overline { \mathrm { XCII } }$$

CDXLIV

Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers and assume $\lim _ { x \rightarrow a } f ( x ) = L$. For a given $\varepsilon \gt 0$, there is one value of $\delta \gt 0$ for which $| f ( x ) - L | \lt \varepsilon$ whenever $0 \lt | x - a | \lt \delta$.

An athletic store sells about 200 pairs of basketball shoes per month when it charges $120 per pair. For each$2 increase in price, the store sells two fewer pairs of shoes. How much should the store charge to maximize monthly revenue? What is the maximum monthly revenue?

Use the extended Euclidean algorithm to find the greatest common divisor of the given numbers and express it as a linear combination of the two numbers. 4158 and 1568

Solve each equation by factoring. $3 x^{2}-x-2=0$