mathematics for economists M1 2015/16: exercise session 16
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integration by parts
indefinite integration by parts
Exercise 232. Verify that
Z
solution:
0
f (x)g (x)dx = f (x)g(x)
f (x)g(x)
Z
0
f (x)g(x)dx + C
0
0
Z
f 0 (x)g(x)dx + C
0
= f

mathematics for economists M1 2015/16: exercise session 5
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composition of functions
Exercise 62. For all the following couples of functions compute and compare
f (g(x) and g(f (x)
p
1. f (x) = x and g(x) = x + 1
p
x 1
and g(x) = x
x+1
p
3

mathematics for economists M1 2015/16: exercise session 10
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theorems on differentiation
the inverse function theorem
Exercise 147. Let f be a function of one independent variable, dierentiable on
(a, b), with nonzero derivative at a point

mathematics for economists M1 2015/16: exercise session 7
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geometric series
Exercise 100. Check that
bn = (1
1
b)(1 + b + b2 + b3 + . + bn 1 )
and use this formula to evaluate the sum
n
X
1
k
b =
solution:
k=0
bn+1
1
b
if b 6= 1 and
n
X
k

mathematics for economists M1 2015/16: exercise session 12
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single variable optimization
extreme values
Exercise 180. Let f be a function of one independent variable. Explain what is
understood under a global maximum of f .
answer: the fu

mathematics for economists M1 2015/16: exercise session 9
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differentiation rules
constant factor derivative rule
Exercise 134. Rewrite each of the following functions as a constant times an
elementary function and find its derivative.
x2

mathematics for economists M1 2015/16: exercise session 11
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approximating functions with polynomials
derivative, tangent line
and
linear approximation
Exercise 163. Find the equation of the tangent line to the graph of a generic
dierentia

mathematics for economists M1 2015/16: exercise session 13
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indefinite integration
the indefinite integral
(or antiderivative)
Exercise 194. Check that
2
F (x) = ex + C,
C 2 R, is an indefinite integral of
2
f (x) = 2xex .
Rewrite the sta

mathematics for economists M1 2015/16: exercise session 8
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derivatives
secant and tangent lines
Exercise 120. Find the slope of the straight line intersecting the graph of the
function f at the points with abscissa a and b.
1. f (x) = x2

mathematics for economists M1 2015/16: exercise session 14
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definite integration
the definite integral
Exercise 206.
(final exam, academic year 2009-2010, 2 points)
Explain what is understood under the definite integral
Z b
f (t)dt.
a
sol

mathematics for economists M1 2015/16: exercise session 17
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integration by substitution
indefinite integration by substitution
Exercise 237. Let F (x) be an indefinite integral of f (x). Verify that
Z
f (g(x)g 0 (x)dx = F (g(x) + C
0
0
0

mathematics for economists M1 2015/16: exercise session 6
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limits
intuitive meaning
Exercise 83. Characterize the behavior of the de-
pendent variable y, when the independent variable
x approaches
y
5
1. a)
2 from the left b)
2 from the r