Negate the following sentences. If f is a polynomial and its degree is greater than 2, then $f^{\prime}$ is not constant.

Each pair of figures is similar. Find the length of x.

Find the domain of the vector-valued function. r(t) = F(t) x G(t), where F(t) = t³i - tj + tk, G(t) = ∛t i + 1 / t+1 j + (t+2)k

Explain why the baking instructions on a box of cake mix are different for high and low elevations. Would you expect to have a longer or a shorter cooking time at a high elevation?

Solve by elimination.

$$\begin{array}{l}{7 x+6 y=33} \\ {2 x-6 y=-6}\end{array}$$

A contractor purchases a bulldozer for $36,500. The bulldozer requires an average expenditure of$11.25 per hour for fuel and maintenance, and the operator is paid $19.50 per hour. (a) Write a linear equation giving the total cost C of operating the bulldozer for t hours. (Include the purchase cost of the bulldozer.) (b) Assuming that customers are charged$80 per hour of bulldozer use, write an equation for the revenue R derived from t hours of use. (c) Use the profit formula (P = R − C) to write an equation for the profit gained from t hours of use. (d) Use the result of part (c) to find the break-even point (the number of hours the bulldozer must be used to gain a profit of 0 dollars).

Graph the curve and draw an arrow specifying the direction corresponding to motion.

$$x = t ^ { 2 } , \quad y = t ^ { 3 }$$

A 13-ft ladder is leaning against a house (see figure) when its base sarts to slide away. By the time the base is 12 ft from the house, the base is moving at the rate of 5ft/sec. How fast is the top of the ladder sliding down the wall at that moment?

This problem is designed to guide you through a “proof” of Plancherel’s theorem, by starting with the theory of ordinary Fourier series on a finite interval, and allowing that interval to expand to infinity. (a) Dirichlet’s theorem says that “any” function f(x) on the interval [-a, +a] can be expanded as a Fourier series: $f(x)=\sum_{n=0}^{\infty}\left[a_{n} \sin \left(\frac{n \pi x}{a}\right)+b_{n} \cos \left(\frac{n \pi x}{a}\right)\right]$. Show that this can be written equivalently as $f(x)=\sum_{n=-\infty}^{\infty} c_{n} e^{i n \pi x / a}$. What is $C_{n}$, in terms of $a_{n}$ and $b_{n} ?$ (b) Show (by appropriate modification of Fourier’s trick) that $c_{n}=\frac{1}{2 a} \int_{-a}^{+a} f(x) e^{-i n \pi x / a} d x$. (c) Eliminate n and $C_{n}$ in favor of the new variables $k=(n \pi / a)$ and $F(k)=\sqrt{2 / \pi} a c_{n}$. Show that (a) and (b) now become $f(x)=\frac{1}{\sqrt{2 \pi}} \sum_{n=-\infty}^{\infty} F(k) e^{i k x} \Delta k ; \quad F(k)=\frac{1}{\sqrt{2 \pi}} \int_{-a}^{+a} f(x) e^{-i k x} d x$, where $\Delta k$ is the increment in k from one n to the next. (d) Take the limit $a \rightarrow \infty$ to obtain Plancherel’s theorem. Comment: In view of their quite different origins, it is surprising (and delightful) that the two formulas—one for F(k) in terms of f(x), the other for f(x) in terms of F(k)—have such a similar structure in the limit $a \rightarrow \infty$.

January 1, 2000, was a Saturday, and 2000 was a leap year. What day of the week will January 1, 2050, be?

Fill in the blank to complete the trigonometric identity. $\cos (-u)=$ ________

Evaluate the expression given that

$$\cos a = \frac { 4 } { 5 } \text { with } 0 < a < \frac { \pi } { 2 }$$

and

$$\sin b = - \frac { 15 } { 17 } \text { with }\frac { 3 \pi } { 2 } < b < 2 \pi.$$

cos(a - b)

Solve each system by elimination.

$$\left\{\begin{array}{l}{x+2 y=-1} \\ {x-y=8}\end{array}\right.$$

Use Newton’s method to find approximate answers to the following questions. Where are all the local extrema of f(x)=$3 x ^ { 4 } + 8 x ^ { 3 } + 12 x ^ { 2 }=+48x$ located?