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Category Archives: Calculus
Taylor Series
Write in first 4 terms of the Taylor series expansion around \(x=0\) for the following functions:
- \(\log(1+x)\)
- \(e^{a x}\)
- \[\frac{1}{1-x}\]
Calculus: Differentiation
Perform the following differentiation:
- \[\frac{d\ }{dx} \Big(x^4\Big) \]
- \[\frac{d\ }{dx} \Big(x^2 \exp(-x)\Big) \]
- \[\frac{d\ }{dx} \Big(\ln(x^2) \Big) \]
Calculus: Infinite Sums
Evaluate the following infinite sums:
- \[ \sum_{k=1}^\infty \big(\frac14\big)^k\]
- \[ \sum_{k=1}^\infty \frac{3^k}{k!}\]
Calculus: Areas
Draw a picture of the region of the \(xy\)-plane were both x and y are between 0 and 1 and \(y \geq x^2\). Find the area of this region
Calculus : Exponentials integrals
Do the following integrals:
- \[ \int_0^1 x^3 dx\]
- (*) Hint: integrate by parts. \[\int_0^\infty x \exp(-x) dx\]
- (**) \[\int_0^\infty \exp(-\frac12 x^2) dx\] Try looking at \[\Big(\int_0^\infty \exp(-\frac12 x^2) dx\Big)\Big(\int_0^\infty \exp(-\frac12 y^2) dy\Big)=\int_0^\infty \int_0^\infty \exp(-\frac12 x^2) \exp(-\frac12 y^2)dx dy\] and then changing to polar coordinates. How des this help you figure out the original question ?
