Calculus

Calculus Warmups

Calculus Warmups: Level 3 Challenges

         

Is f(x)=x+sinxf(x)= x + \sin x an injective function?

j=0i=02(i+j)i+j+1= ?\large\sum _{ j=0 }^{ \infty }{ \sum _{ i=0 }^{ \infty }{ \frac { { 2 }^{ -(i+j) } }{ i+j+1 } } } = \ ?

Give your answer to 3 decimal places.

limxy1(lnxlny+lnylnx)= ? \large \lim _{xy\to 1}\left(\frac{\ln\:x}{\ln\:y}+\frac{\ln\:y}{\ln\:x}\right)= \ ?

2×limh011+hxxdx1h1xxdxh2=? \large 2 \times \lim_{h\to0} \dfrac{\int_1^{1+h} x^x \, dx - \int_{1-h}^1 x^x \, dx }{h^2} =\, ?

Some people claim that 00=1 0 ^ 0 = 1 . What is

limx0+x1lnx? \Large \lim_{ x \rightarrow 0^+ } x ^{^ { \frac{ 1}{\ln x} }}?

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