We frequently must solve equations of the form f(x) = 0. When f is a continuous function on [a, b] and f(a) and f(b) have opposite signs, the intermediate-value theorem guarantees that there exists at least one solution of the equation f(x) = 0 in [a, b]. (a) Explain in words why there exists exactly one solution in (a, b) if, in addition, f is differentiable in (a, b) and $f^{\prime}(x)<0$ is either strictly positive or strictly negative throughout (a, b). (b) Use the result in (a) to show that $x^{3}-4 x+1=0$ has exactly one solution in [−1, 1].

Distance In Exercises $75-80$, find the distance between the point and line, or between the lines, using the formula for the distance between the point $\left(x_1, y_1\right)$ and the line $A x+B y+$ $C=0$

$$\text { Distance }=\frac{\left|A x_1+B y_1+C\right|}{\sqrt{A^2+B^2}}$$

Point: $(-2,1)$ Line: $x-y-2=0$

A brass rod $12.0$ cm long, a copper rod $18.0$ cm long, and an aluminum rod $24.0$ cm long --- each with cross-sectional area $2.30$ cm$^3$ ---are welded together end to end to form a rod $54.0$ cm long, with copper as the middle section. The free end of the brass section is maintained at $100.0{\degree}C$, and the free end of the alumi- num section is maintained at $0.0{\degree}C$. Assume that there is no heat loss from the curved surfaces and that the steady-state heat current has been established. What is
(b) the temperature $T_2$ at the junction of the copper and aluminum sections;

At a certain distance from the center of the Earth, a $4.6 \mathrm{~kg}$ object has a weight of $2.2 \mathrm{~N}$. Find this distance.

For items given, fill in the missing letters in each word. Unscramble the letters in the boxes to find the answer to item 6 .

Les p _ _ _ _ - $\square$ de David sont arrivés des États-Unis.

A long iron rod $\left(\rho=7870 \mathrm{kg} / \mathrm{m}^{3}, c_{p}=447 \mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\right., k=80.2 \mathrm{W} / \mathrm{m} \cdot \mathrm{K},$ and $\alpha=23.1 \times 10^{-6} \mathrm{m}^{2} / \mathrm{s} )$ with diameter of 25 mm is initially heated to a uniform temperature of $700^\circ C.$ The iron rod is then quenched in a large water bath that is maintained at constant temperature of $50^\circ C$ and convection heat transfer coefficient of $128 W/m^2 \cdot K.$ Determine the time required for the iron rod surface temperature to cool to $200^\circ C.$

In each of the following sentences, underline each direct object once and each indirect object twice. Some sentences contain compound objects.

According to our guide, that diamond caused its owner some trouble.

A bird is flying at an elevation of 14 feet above the surface of the water. A fish is swimming the same distance below the surface ofthe water. a. What number represents the position of the fish relative to the surface ofthe water? b. How does the absolute value ofthe number you wrote show that the distances are the same? Explain.

Suppose p is a positive integer such that the function f is p-times differentiable and $f^{(p)}=f$. Using mathematical induction, show that f is in fact n-times differentiable for every positive integer n and that each of its higher derivatives $f^{(n)}$ equals one of the p functions $f, f^{\prime}, f^{\prime \prime}, \ldots, f^{(p-1)}$.

Use the Distance Formula to find the distance between each pair of points. C (11, -12), D (6, 2)

Answer the following questions in complete sentences:

1. ¿Saca Sarita una mala nota en inglés?
2. ¿Está ocupada Sandra? Explica. 3.¿Está David deprimido? Explica.

In the space above each sentence below, add at least one adverb. Use a caret ($\land$) to mark where the adverbs are inserted. The caravan arrived at the oasis, and everyone helped to set up camp.

A small object is placed $20 \ \mathrm{cm}$ from the first of a train of three lenses with focal lengths, in order, of $10$, $15$, and $20\ \mathrm{cm}$.. The first two lenses are separated by $30 \ \mathrm{cm}$ and the last two by $20 \ \mathrm{cm}$ . Calculate the final image position relative to the last lens and its linear magnification relative to the original object when (a) all three lenses are positive, (b) the middle lens is negative, (c) the first and last lenses are negative. Provide ray diagrams for each case.

For a random variable that is normally distributed, with $\mu=200$ and $\sigma=20$, determine the probability that a simple random sample of 4 items will have a mean that is

a. greater than $210 .$