\[ \ln(x) = \int\limits_1^x \frac{1}{t}\,\mathrm{d}t \qquad \int\limits \frac{1}{x}\,\mathrm{d}x = \ln|x| + C \qquad \frac{\mathrm{d}}{\mathrm{d}x}\ln|x| = \frac{1}{x} \]
- Using this “geometric” definition of \(\ln(x)\) as an integral, show that the number \(\ln(2)\) must be between ½ and ¾. Can you come up with tighter bounds?
- Calculate the derivatives of the following functions explicitly: \[ f(x) = \ln(7x^2) \qquad g(x) = \ln\big(\cos^2(x)\big) \qquad h(x) = x\ln\!\big(\ln(x)\big) \]
- Calculate the exact values of the following definite integrals: \[ \int\limits_0^{\pi/4} \tan(x) \,\mathrm{d}x \qquad \int\limits_1^\mathrm{e} \frac{\ln(x)}{x} \,\mathrm{d}x \qquad \int\limits_2^3 \frac{x^2}{x^3-5} \,\mathrm{d}x \]