Exam 6: Applications of Integration

A lighthouse is located on a small island, $3 \mathrm {~km}$ away from the nearest point $P$ on a straight shoreline, and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is $1 \mathrm {~km}$ from $P$ ?

Differentiate the function. $y = x ^ { 6 / x }$

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

Find the given integral. $\int \frac { 2 \tan ^ { - 1 } 2 x } { 1 + 4 x ^ { 2 } } d x$

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

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

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?

Find an equation of the tangent line to the curve at the given point. $y = 6 e ^ { 2 x } \cos \pi x , ( 0,6 )$

Calculate $g ( x )$ , where $g = f ^ { - 1 }$ . State the domain and range of $g$ . Calculate $g ^ { \prime } ( a )$ $f ( x ) = \frac { 1 } { x - 3 } , x > 3 ; a = 2$

Find the inverse of the function. $f ( x ) = \frac { 1 + 8 x } { 8 - 5 x }$

If $f ( x ) = 8 x + \ln x$ , find $f ^ { - 1 } ( 8 )$

Use logarithmic differentiation to find the derivative of the function. $y = ( x + 1 ) ^ { 2 } \left( 8 x ^ { 2 } - 5 \right) ^ { 3 }$

Find the solution of the equation correct to four decimal places. $\ln \left( e ^ { x } - 3 \right) = 2$

Find the derivative of the function. $y = 3 \sin ^ { - 1 } \left( x ^ { 2 } \right)$

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

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

Find the volume of the solid obtained by rotating about the y axis the region bounded by the curves. $y = e ^ { - x ^ { 2 } } , y = 0 , x = 0$ and $x = 9$

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 logarithmic differentiation to find the derivative of the function. $y = ( x + 2 ) ^ { 7 / x }$

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