# Practice Skills Test 1
#  Instructions:   This is a practice exam for the first 'skills' test
# to be given on Tuesday,  September
# 23.   You can use any calculator  up to,  but not including, a TI-92. 
#   You must show your
# work. 
# \medskip
# 1.    Suppose   Int(f(x),x=1..5)=1.2 ,  and  Int(g(x),x=1..5)=-3.1.  
# Evaluate the integral
# Int(3*x+1+4*f(x) - g(x),x=1..5).
# 
# \vfill
# 
# 
# 
# 2.    Suppose  the temperature rose linearly from 60 degrees to 80
# degrees from 9 am to 12 am, and then stayed at 80 degrees for the next
# two hours.  Calculate the average  temperature over the 5   hour time
# interval.   
# 
# \vfill
# 
# 
# 
# 3.   Evaluate Int(f(x),x=0..4), where  y=f(x) is the function graphed
# below.
> plot([[0,-2],[3,4],[4,3]]);

# \newpage
# 4.    Suppose f  is a continuous function with f(x)>0 for  x<5 and 
# f(x) <0 for x>5.
# Let A[f](x)=Int(f(t),t=2..x.      Make a rough sketch of the graph of
# A[f]over the interval  0 to 8, showing where  A[f]  is increasing and
# where it is decreasing.
# 
# \vfill
# 
# 5.    Differentiate the following functions.  
# a)     ln(a*x+b)
# 
# \vfill
# b)    arcsin(x^2+1 
# 
# \vfill
# 
# 
# c)   exp(3*x+1)*sin(2*x)  
# 
# \vfill
# 
# 6.   Let  I=Int(sqrt(x+10),x=0..5).   Write out the  approximation R[5
#  to I..     Is R[5]>I?  Why or why not?
# \vfill
# 
# ~
# \newpage
# 
# 7.    The velocity of a particle moving on a straight line was
# measured at  times 
# t=0,1,3,4  (measured in seconds) and found to be  respectively
# v=20,10,-5,-10.
# Estimate, on the information given, the average velocity and the
# average speed over the time interval  from  t=0..7.
# 
# \vfill
# 
# 8.   Find the area between the functions  y=x^2 -4 and  y=4-x^2.  
# Sketch the region.
# 
# \vfill
# 
# ~
# \newpage
# 9.   Perform the following integrations:
# 
#  
# a)   Int(x*sin(x^2),x)                           
# 
# \vfill
# b)   Int(x/sqrt(1-x^2),x)
# 
# \vfill
# c)   Int(x/(3*x+1),x=1..5)
# 
# \vfill
# d)   Int(3x^3/(1+x^4),x)
# 
# \vfill
#  10.    Given the error bound   abs(Int(f(x),x=a..b)-L[n]) <=
# (K[1]*(b-a)^2)/(2*n)   for the left hand rule.
#  Take f(x)= x^2  and a=0,  b=10.     What value of n do we need in
# order that
# abs(Int(x^2,x=0..1)-L[n]) be less or equal to 1.0?
# \vfill
# ~