Boats $$A$$ and $$B$$ leave the same place at the same time. Boat $$A$$ heads due north at $$12\frac{\text{km}}{\text{hr}}$$ . Boat $$B$$ heads due east at $$18\frac{\text{km}}{\text{hr}}$$ . After $$2.5$$ hours, how fast is the distance between the boats increasing (in $$\frac{\text{km}}{\text{hr}}$$ )?

$$\lim_{h\rightarrow0}\frac{\tan\left(\frac{\pi}{6}+h\right)-\tan\left(\frac{\pi}{6}\right)}{h}=$$

If $$\int_{30}^{100}f\left(x\right)dx=A$$ and $$\int_{50}^{100}f\left(x\right)dx=B$$ , then $$\int_{30}^{50}f\left(x\right)dx=$$

The graph of $$y=x^3-5x^2+4x+2$$ has a local minimum at

The volume generated by revolving about the $$y$$ -axis the region enclosed by the graphs $$y=9-x^2$$ and $$y=9-3x$$ , for $$0\le x\le2$$ , is

The average value of the function $$f\left(x\right)=\left(\ln x\right)^2$$ on the interval $$\left[2,4\right]$$ is

$$\frac{d}{dx}\left(\int_0^{3x}\cos t\ dt\right)=$$  

Error is defined as the positive difference between an actual value and an approximation. If the definite integral $$\int_1^3\left(x^2+1\right)dx$$ is approximated by using a trapezoid sum with $$4$$ trapezoids, the error is

The radius of a sphere is increasing at a rate proportional to itself. If the radius is $$4$$ initially, and the radius is $$10$$ after $$2$$ seconds, what will the radius be after $$3$$ seconds?

Approximate the change in the volume of a sphere when the radius is increased from $$10$$ to $$10.02\ \text{cm}$$ .

$$\int_{ }^{ }\ln2x\ dx=$$  

If the function $$f\left(x\right)$$ is differentiable, and:

$$f\left(x\right)=ax^3-6x$$ if $$x\le1$$ ,

$$f\left(x\right)=bx^2+4$$ if $$x>1$$ ,

then $$a=$$

Two particles leave the origin at the same time and move along the $$y$$ -axis with their respective positions determined by the functions $$y_1=\cos2t$$ and $$y_2=4\sin t$$ for $$0<t<6$$ . For how many values of $$t$$ do the particles have the same acceleration?

Find the distance traveled in the first four seconds for a particle whose velocity is given by $$v\left(t\right)=7e^{-t^2}$$ where $$t$$ stands for time in seconds.

$$\int_{ }^{ }\tan^6x\sec^2x\ dx=$$