Problem 65
Evaluate the Riemann sums \(L_{4}\) and \(R_{4}\) for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer. \(y=3 x^{2}-2 x+1\) over \([-1,1]\)
Problem 65
The following exercises are intended to derive the fundamental properties of the natural log starting from the definition ln(x)=?x1dtt, using properties of the definite integral and making no further assumptions. Use a change of variable in the integral \(\int_{1}^{x y} \frac{1}{t} d t\) to show that \(\ln x y=\ln x+\ln y\) for \(x, y>0\).
Problem 66
Evaluate the Riemann sums \(L_{4}\) and \(R_{4}\) for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer. \(y=\ln \left(x^{2}+1\right)\) over \([0, e]\)
Problem 66
The following exercises are intended to derive the fundamental properties of the natural log starting from the definition ln(x)=?x1dtt, using properties of the definite integral and making no further assumptions. Use the identity \(\ln x=\int_{1}^{x} \frac{d t}{x}\) to show that \(\ln (x)\) is an increasing function of \(x\) on \([0, \infty)\), and use the previous exercises to show that the range of \(\ln (x)\) is \((-\infty, \infty) .\) Without any further assumptions, conclude that \(\ln (x)\) has an inverse function defined on \((-\infty, \infty)\).
Problem 67
Prove the expression for \(\sinh ^{-1}(x)\). Multiply \(x=\sinh (y)=(1 / 2)\left(e^{y}-e^{-y}\right)\) by \(2 e^{y}\) and solve for \(y\). Does your expression match the textbook?
Problem 67
The following exercises are intended to derive the fundamental properties of the natural log starting from the definition ln(x)=?x1dtt, using properties of the definite integral and making no further assumptions. Pretend, for the moment, that we do not know that \(e^{x}\) is the inverse function of \(\ln (x)\), but keep in mind that \(\ln (x)\) has an inverse function defined on \((-\infty, \infty)\). Call it \(E\). Use the identity \(\ln x y=\ln x+\ln y\) to deduce that \(E(a+b)=E(a) E(b)\) for any real numbers \(a, b\).
Problem 67
Evaluate the Riemann sums \(L_{4}\) and \(R_{4}\) for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer. \(y=x^{2} \sin x\) over \([0, \pi]\)
Problem 68
Evaluate the Riemann sums \(L_{4}\) and \(R_{4}\) for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer. \(y=\sqrt{x}+\frac{1}{x}\) over \([1,4]\)
Problem 68
Prove the expression for \(\cosh ^{-1}(x)\). Multiply \(x=\cosh (y)=(1 / 2)\left(e^{y}-e^{-y}\right)\) by \(2 e^{y}\) and solve for \(y .\) Does your expression match the textbook?
Problem 69
The following exercises are intended to derive the fundamental properties of the natural log starting from the definition ln(x)=?x1dtt, using properties of the definite integral and making no further assumptions. The sine integral, defined as \(S(x)=\int_{0}^{x} \frac{\sin t}{t} d t\) is an important quantity in engineering. Although it does not have a simple closed formula, it is possible to estimate its behavior for large \(x .\) Show that for \(k \geq 1,|S(2 \pi k)-S(2 \pi(k+1))| \leq \frac{1}{k(2 k+1) \pi} \cdot(\operatorname{Hin} t: \sin (t+\pi)=-\sin t)\)
