Exam 6: Applications of Integration

Find the exact value of the given expression. $\sin ^ { - 1 } \frac { 1 } { 2 }$

The geologist C. F. Richter defined the magnitude of an earthquake to be $\log _ { 10 } \left( \frac { I } { S } \right)$ where $I$ is the intensity of the quake (measured by the amplitude of a seismograph $100 \mathrm {~km}$ from the epicenter) and $S$ is the intensity of a "standard" earthquake (where the amplitude is only 1 micron $= 10 ^ { - 4 }$ $\mathrm { cm } )$ . The 1989 Loma Prieta earthquake that shook San Francisco had a magnitude of $7.9$ on the Richter scale. The 1906 San Francisco earthquake was 12 times as intense. What was its magnitude on the Richter scale?

Use transformations to sketch the graph of the function. $y = 3 \ln ( x - 5 )$

Determine $f ( x )$ from the table. The values from the table are given to the nearest ten thousandth. x f(x) -1 20.0855 0 1 1 0.0498 2 0.0025

Suppose that the graph of $y = \log _ { 2 } x$ is drawn on a coordinate grid where the unit of measurement is an inch. How many miles to the right of the origin do we have to move before the height of the curve reaches $2 \mathrm { ft }$ .

Use the laws of logarithms to write the expression as the logarithm of a single quantity. $4 \ln 3 - \frac { 3 } { 4 } \ln ( x + 3 )$

Use the laws of logarithms to expand the expression. $\ln \left( \frac { x + 5 } { x - 6 } \right) ^ { 1 / 2 }$

Determine whether the function is one-to-one. Select the correct answer. $f ( x ) = \sqrt { 3 - x }$

Differentiate the function. $y = \ln \left( x ^ { 3 } \sin ^ { 2 } x \right)$

Differentiate the function. $y = x \left( 5 ^ { 2 x } \right)$

Use the properties of logarithms to expand the quantity. $\log _ { 2 } \left( \frac { x ^ { 10 } y } { z ^ { 5 } } \right)$

Evaluate the integral. $\int _ { 0 } ^ { 2 } \frac { e ^ { x } } { 1 + e ^ { 2 x } } d x$

Find the limit. $\lim _ { x \rightarrow \infty } e ^ { 4 - x ^ { 4 } }$

Solve each equation for $x$ . (a) $\ln x = 4$ (b) $e ^ { e ^ { x } } = 7$

Find the integral. $\int ( x + 1 ) 2 ^ { x ^ { 2 } + 2 x } d x$

Find the derivative of the function. $y = \left( 4 x ^ { 2 } + 1 \right) \tan ^ { - 1 } 2 x$

Use the graph of $y = \ln x$ as an aid to sketch the graph of the function. $g ( x ) = \ln ( x + 1 )$

Find the limit. $\lim _ { x \rightarrow 2 ^ { - } } \log _ { 15 } \left( x ^ { 2 } - 5 x + 2 \right)$

Find the derivative of the function. $y = \left( 4 x ^ { 2 } + 1 \right) \tan ^ { - 1 } 2 x$