Slope Fields Use a computer algebra system to graph the slope field for the differential equation and graph the solution satisfying the specified initial condition.

$$\frac{dy}{dx}=\frac{10}{x \sqrt{x^2-1}},\ y(3)=0$$

Find the integral.

$$\displaystyle\int \frac{t}{\sqrt{1-t^4}} \ dt$$

Determine the limit of the trigonometric function (if it exists).

$$lim_{t\to0} \ \frac{\sin 3t}{2t}$$

Prove that

$$\arctan x+\arctan y = \arctan \frac{x+y}{1-xy}, \ xy\neq1$$

Find the indefinite integral.

$$\displaystyle\int (2x-9\sin x)\ dx$$

Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution.

$$\frac{dy}{dx}=\frac{1}{x\sqrt{x^2+4}},\ (2,1)$$

Slope Fields A differential equation, a point, and a slope field are given. Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the given point.

$$\frac{dy}{dx}=\frac{3}{1+x^2},\quad (0,0)$$

Length Rework the problem below, given that one hallway is of width $a$ meters and the other is of width $b$ meters.

A hallway of width $6$ feet meets a hallway of width $9$ feet at right angles. Find the length of the longest pipe that can be carried level around this corner. [Hint: If $L$ is the length of the pipe, show that

$$L=6\csc \theta+9\csc\left(\frac{\pi}{2}-0\right)$$

where \theta is the angle between the pipe and the wall of the narrower hallway.]

Find the derivative of the function.

$$y=\cosh^{-1} (3x)$$

Determine the limit of the trigonometric function (if it exists).

$$lim_{x\to\pi/4} \ \frac{1-\tan x}{\sin x- \cos x}$$

Use a computer algebra system to find or evaluate the integral. Graph the integrand.

$$\int_{-\pi/4}^{\pi/4} \frac{sin^2 x -\cos^2 x}{\cos x} \ dx$$

Find the derivative of the function.

$$y= \sqrt x + \frac{1}{4} \sin (2x)^2$$

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

The range of $y=\arcsin x$ is $[0,\pi]$

Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution.

$$\frac{dy}{dx}=\frac{3}{1+x^2},\ (2,1)$$