Limits and Derivatives
\(\frac{d}{dx}(xe^x+\sin x)\)
What is \(e^x+xe^x+\cos x\)?
\(\displaystyle\int_{-1}^1\csc^2x\textrm{d}x\)
What is unsolvable with what we currently know since there is a discontinuity at 0?
The area under a sine curve from 0 to \(\frac{\pi}{2}\)
What is 1?
A particle is moving with a position defined by \(s(t)=e^t+t\). Find the acceleration of the particle at time \(t=2\).
What is \(e^2\)?
\[\lim_{x\to 1}\frac{e^{x-1}-1}{\frac{1}{x}-1}\]
What is -1?
\(\int\sin^2x\textrm{d}x\)
What is \(\frac{1}{2}x-\frac{1}{4}\sin(2x)+c\)?
The area between \(y=x^2\) and \(y=\frac{1}{x^2+1}\) from \(x=\frac{1}{\sqrt{3}}\) to \(\frac{1}{\sqrt{3}}\).
What is \(\frac{\pi}{3}-\frac{2\sqrt{3}}{27}\)?
The tangent line at \(x=\frac{1}{2}\) to the top half of the unit circle.
What is \(y=\frac{-\sqrt{3}}{3}\left(x-\frac{1}{2}\right)+\frac{\sqrt{3}}{2}\)?
Let \(f(2)=0\), \(f'(2)=-1\), \(g(0)=1\), \(g(2)=-1\), \(g'(2)=5\), and \(g'(0)=4\). Also let \(h(x)=\frac{\textrm{d}}{\textrm{d}x}\left[g\left(f(x)\right)\cdot g(x)\right]\). Find \(h(2)\).
What is 9?
\(\displaystyle\int_{\frac{1}{2}}^1\sqrt{1-x^2}\textrm{d}x\)
(Hint: Geometry)
What is \(\frac{\pi}{6}-\frac{\sqrt{3}}{8}\)?
The volume of the solid generated by rotating the region bounded by the graph of \(y=4-x^2\) and the \(x\)-axis about the line \(y=4\).
What is \(\frac{1792\pi}{15}\)?
Mr. Garret is driving directly eastward at 2 units per second. Dr. Morris is driving 4 units per second at a \(30^\circ\) angle North of East. They both start at the origin. How fast is the distance between the two vehicles changing when they are \(2\sqrt{3}\) units apart? (i.e. \(t=1\)).
What is \(2\sqrt{3}\) units per second?
\(\frac{\textrm{d}}{\textrm{d}x}x^x\) when \(x=2\).
\(4+4\ln 2\)
Daily Double
What is \(-\frac{1}{8}\left(\ln 2\right)^2\)?
(See Presentation)
What is \(\(frac{2}{3}\left(\ln 4\right)^\frac{3}{2}\)?
The maximum area of a rectangle (parallel to the \(x\)- and \(y\)-axes) with one corner at the origin and the other on the curve \(y=\frac{1}{1+x^2}\).
What is \(\frac{1}{2}\)?
\(\displaystyle\lim_{x\to0^+}\left(\cos x\right)^{\frac{1}{x^2}}\)
What is \(\frac{1}{2}\)?
What is 0?
The volume when the region between \(x=2-y^2\) and \(x=4-2y^2\) is revolved about the \(y\)-axis.
What is \(2\pi\)?
Daily Double
What is \(\frac{-11\sqrt{3}\pi}{30\sqrt{13}}\)?