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practice2

# practice2 - 9 Look at the picture in problem 7 on the test...

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Practice problems 1. If f and g are inverse functions of each other, show that g ( x ) = 1 /f ( g ( x )). (Note: The symbol f ( g ( x )) on the right side means that in the expression for f ( x ), one substitutes g ( x ) instead of x .) Use this formula to derive the formula for derivative of arccsc x. 2. Differentiate ( a ) f ( x ) = e e x 2 , ( b ) f ( x ) = arctan ( e x 3 ) 3. Differentiate ( a ) f ( x ) = arcsin ( ln ( x 3 )) , ( b ) f ( x ) = arcsin ( ln 3 ( x )) 4. Differentiate ( a ) f ( x ) = arccos ( sec 2 ( x 3 )) , ( b ) f ( x ) = arccsc x cos ( x 2 ) 5. Differentiate ( a ) f ( x ) = x 4 4 x , ( b ) f ( x ) = 5 - (1 /x ) (page 475, problems 24 and 25) 6. Differentiate ( a ) f ( x ) = 10 tan x , ( b ) f ( x ) = ( ln x ) x (page 475, problems 26 and 35) 7. Given an ellipse x 2 + xy + y 2 = 3, find (a) dy/dx, (b) tangent line to the ellipse at the point (1 , 1). (page 189, problem 25) 8. A particle is moving along the curve y = x . As the particle passes through the point (4 , 2) , its x-coordinate increases at a rate of 3 cm/s . How fast is the distance from the particle to the origin changing at that instant? (page 203, problem 18)
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Unformatted text preview: 9. Look at the picture in problem 7 on the test 1. Label the corners of the shaded triangle as A , B , C , such that the point A is the point where the triangle touches the curve f ( x ) = x 2 , point B is the one directly below A and point C = (2 , 0) is ﬁxed. If the point be B starts at the origin and moves to the right with the speed of 1 cm s (assume all the lengths on the picture are in cm ), how fast is the angle 6 ( BAC ) changing a) when x = 1? b) when the area of the triangle ABC is maximal? 10. Two sides of a triangle have lengths 5 cm and 4 cm . The angle between them is increasing at a constant rate of 0 . 05 rad/h . Find the angle at which the rate of change of the third side is maximal....
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