Exercises for 1st and 2nd Day Review

 


Find the derivatives of the following functions. Check your work using symbolic differentiation on the TI-89.

1.  f( x) 
= 
x2+sin( 2x)         2.  f(x) = ex+e-2x        3.  f( x) = ln æ
ç
è		 1
x2+1		 ö
÷
ø
4.  f( x) 
= 
tan2( x2)            5.  f( x) = ecos( x)         6.  f(x) = x2cot( x3) 

Find the antiderivatives of the following functions. Check your work using symbolic differentiation on the TI-89.

7.  ó
õ		 ( x2+ex) dx        8.  ó
õ		 xcos(x2) dx        9.  ó
õ		 x2+1
x		   dx
10.  ó
õ		 ( cos2( x) -sin2( x)) dx        11.  ó
õ		 ecos( x) sin(x) dx

Find the derivative of the following functions using logarithmic differentiation. Check your work using symbolic differentiation on the TI-89.

12.  y =  x2ex
x3+1		             13.  y =  æ
ç
è		 x3
x5+ex		 ö
÷
ø		 10
 
 

Find the antiderivatives using integration by parts. Check your work using symbolic differentiation on the TI-89.

14.  ó
õ		 xsin( x) dx        15.  ó
õ		 x2e2xdx       16.  ó
õ		 xln( x) dx
17.  ó
õ		 xcos( 3x) dx        18.  ó
õ		 xsec2(x) dx

Solve the following separable equations:

19.  xy' = y2        20.  yy' = y2+1       21.  y' = xy+x

Be prepared to answer the following discussion questions:

22. What is a definite integral?

23. What does a definite integral represent when the integrand is positive?

24. What is the midpoint approximation of a definite integral over a given partition of the interval of integration?