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Test 2 practice problems

Test 2 practice problems - Practice Problems for Test 2...

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Practice Problems for Test 2 Note: These are not sample test questions. However, your understanding of the topics below will help you on your second test. You should also review the derivatives on Worksheet 5 and the concepts presented in Worksheets 3, 4, and 6. Be sure to review all class notes and assigned homework problems. 1) Use the definition: ! f ( x ) = lim h " 0 f ( x + h ) # f ( x ) h to show that ! f ( x ) = 4 x " 3 if f ( x ) = 2 x 2 ! 3 x 2) Use g ( x ) = x + 1 if x " 2 2 x # 1 if x > 2 $ % & a) Is g ( x ) continuous for all x ? b) Is g ( x ) differentiable for all x ? How did you decide? c) Sketch the graphs of g ( x ) and ! g ( x ) . 3) Find dy dt for each of the following: a) y = t 3 cos 3 ( t 3 ) b) sin 2 (5 t ) + cos 2 (3 y ) = 2 t c) y = 10 t ! 5 d) y = ( t 2 + 5) 10 (1 ! t ) e) y = sin( ! 2 t 2 ) f) y = t 3 tan( t ! 1) g) y = sin(cos t ) h) y = cos( ! ) i) y = tan ! 1 (cos t ) j) y = e 5 t 1 + 3 t k) y = 5 4 t 3 l) sin y = sin ! 1 t 4) Find the equation for the line that is tangent to the graph of the given function, y = f ( x ) , at the given point: a) f ( x ) = sin(5 x ) at the point ( ! , f ( ! )) b) x 2 y 3 ! 4 y = 7 x ! 4 at the point (1, ! 1) c) y = x ( x 2 + 1) 5 at the point 1, 1 32 ( ) 5) If x = r 2 ! 2 and r = cos( " ) find dx d !
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6) Suppose y and z are both differentiable functions of t and suppose y (3) = 2 and
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