The height h in feet of a car on a different Ferris wheel can be modeled by $h(t)=-16 \cos \frac{\pi}{45} t+24$, where t is the time in seconds. Determine the period, midline, amplitude, and maximum and minimum values of the graph. For one cycle starting from t=0, identify all points where the graph intersects its midline and the coordinates of any local maxima and minima. Interpret these points in the context of the problem, and graph one cycle.

The derivative $f^{\prime}(x)$ of a differentiable function $f(x)$ is given. In each case,

(a) Find intervals of increase and decrease for $f(x)$.

(b) Determine values of $x$ for which relative maxima and minima occur on the graph of $f(x)$.

(c) Find $f^{\prime \prime}(x)$ and determine intervals of concavity for the graph of $f(x)$.

(d) For what values of $x$ do inflection points occur on the graph of $f(x)$ ?

Complete each of the following.

a. Graph the function by making a table of values.

b. Determine the consecutive values of $x$ between which each real zero is located.

c. Estimate the $x$-coordinates at which the relative maxima and minima occur.

$$f(x)=x^4+4 x^3+6 x^2+4 x-3$$

$$\begin{array}{|l|l|} \hline x & f(x) \\ \hline & \\ \hline & \\ \hline & \\ \hline & \\ \hline & \\ \hline \end{array}$$

The derivative $f^{\prime}(x)$ of a differentiable function $f(x)$ is given. In each case,

(a) Find intervals of increase and decrease for $f(x)$.

(b) Determine values of $x$ for which relative maxima and minima occur on the graph of $f(x)$.

(c) Find $f^{\prime \prime}(x)$ and determine intervals of concavity for the graph of $f(x)$.

(d) For what values of $x$ do inflection points occur on the graph of $f(x)$ ?

$$f^{\prime}(x)=x^3(x-2)^2$$

A function and its first and second derivatives are given. Use these to find any horizontal and vertical asymptotes, critical points, relative maxima, relative minima, and points of inflection. Then sketch the graph of each function.

$$y=3 \sqrt[3]{x}+\frac{1}{x}$$

$$y^{\prime}=\frac{x^{4 / 3}\ -\ 1}{x^2}$$

$$y^{\prime \prime}=\frac{6\ -\ 2 x^{4 / 3}}{3 x^3}$$

A function and its first and second derivatives are given. Use these to find any horizontal and vertical asymptotes, critical points, relative maxima, relative minima, and points of inflection. Then sketch the graph of each function.

$$f(x)=\frac{3 x^{2 / 3}}{x\ +\ 1}$$

$$f^{\prime}(x)=\frac{2\ -\ x}{x^{1 / 3}(x\ +\ 1)^2}$$

$$f^{\prime \prime}(x)=\frac{2\left(2 x^2-8 x-1\right)}{3 x^{4 / 3}(x+1)^3}$$

Una moto con las ruedas desinfladas de 45.0 kg de masa descansa sobre un piso horizontal. Usted comienza a empujarla hasta la gomería, ejerciendo sobre ella un empuje horizontal cada vez mayor, hasta que observa que la moto empieza a moverse. El coeficiente de fricción estática entre la moto y la superficie es de 0.30, y el coeficiente de fricción cinemática es de 0.10. ¿Qué fuerza horizontal mínima deberá aplicar usted para que la moto siga moviéndose con velocidad constante, una vez que haya comenzado a moverse?

A function and its first and second derivatives are given. Use these to find any horizontal and vertical asymptotes, critical points, relative maxima, relative minima, and points of inflection. Then sketch the graph of each function.

$$f(x)=\frac{9(x\ -\ 2)^{2 / 3}}{x^2}$$

$$f^{\prime}(x)=\frac{12(3\ -\ x)}{x^3(x\ - \ 2)^{1 / 3}}$$

$$f^{\prime \prime}(x)=\frac{4\left(7 x^2\ -\ 42 x\ + \ 54\right)}{x^4(x\ - \ 2)^{4 / 3}}$$

Un alambre de $b \mathrm{~m}$ de largo se corta en dos pedazos. Una pieza se dobla en un triángulo equilátero y la otra se dobla en un círculo. Si la suma de las áreas encerradas por cada parte es mínima, ¿cuál es la longitud de cada parte?

Show that A is a scalar matrix kI if and only if the minimal polynomial of A is m(t)=t-k.

Identify each statement as true or false. If a statement is false, explain why. a. The relative maxima of the graph of $y=\sin x$ correspond to the relative minima of the graph of $y=\csc x$. b. The period of $y=\tan 2 x$ is $\pi$. c. The amplitude of $y=2 \cot x$ is 2 . d. The period of $y=\sec 2 x$ is $\pi$. e. The vertical asymptotes of the graph of $y=\cot x$ occur where the graph of $y=\cos x$ has $x$-intercepts.

The derivative f'(x) of a differentiable function f(x) is given. (a) Find intervals of increase and decrease for f(x). (b) Determine values of x for which relative maxima and minima occur on the graph of f(x). (c) Find f''(x), and determine intervals of concavity for the graph of f(x). (d) For what values of x do inflection points occur on the graph of f(x)? $f^{\prime}(x)=\frac{x+3}{(x-2)^{2}}$

The derivative f'(x) of a differentiable function f(x) is given. (a) Find intervals of increase and decrease for f(x). (b) Determine values of x for which relative maxima and minima occur on the graph of f(x). (c) Find f''(x), and determine intervals of concavity for the graph of f(x). (d) For what values of x do inflection points occur on the graph of f(x)? $f^{\prime}(x)=\frac{x+2}{(x-1)^{2}}$

The derivative $f^{\prime}(x)$ of a differentiable function $f(x)$ is given. In each case,

(a) Find intervals of increase and decrease for $f(x)$.

(b) Determine values of $x$ for which relative maxima and minima occur on the graph of $f(x)$.

(c) Find $f^{\prime \prime}(x)$ and determine intervals of concavity for the graph of $f(x)$.

(d) For what values of $x$ do inflection points occur on the graph of $f(x)$ ?

$$f^{\prime}(x)=x^2(x+1)^3$$