MAA151 Single Variable Calculus academic year 2016/17
Exercises in class A – Set 1 (5) – Functions
1.
2.
√
Find the inverse of the function f defined by f (x) = −1/ x − 1. Especially,
specify the domain and the range of the inverse. Also, sketch in the same figure
the graphs of f and f −1 .
√
Let f (x) = x + 1, g(x) = ex and h(x) = ln(x + 3). Find the composition
f ◦ g ◦ h, and specify its domain and range.
3.
Let f (x) = |x|. Explain and illustrate how the curve given by the equation
y + 3 = f (x − 2) may be obtained from the curve given by y = f (x).
4.
Determine whether the function f definied by f (x) = x/(x + 1), x > −1 is
non-decreasing, increasing, non-increasing, decreasing or neither of those four
alternatives.
5.
Determine for each of the functions f1 , f2 , f3 , f4 , definied by
f1 (x) = e−x ,
f2 (x) =
x(x + 1)
,
x2 + 1
√
f3 (x) = 1/ x ,
f4 (x) = sinh(x) ,
whether it is bounded, bounded above, bounded below or (completely) unbounded.
6.
Let f (x) = sin(x). Explain and illustrate how the curve given by the equation
y = 2f (4x/3) may be obtained from the graph of f .
MAA151 Single Variable Calculus academic year 2016/17
Exercises in class A – Set 2 (5) – Limits, continuity, geometric series
1.
Find the limits.
a)
b)
c)
d)
lim
x→4
x2 + 2x − 24
x2 − 7x + 12
e)
sin(3x)
lim
x→0 |x + 1| − |x − 1|
f)
√
x + x2 − x
g)
lim
x→+∞
h)
lim x sin(1/x)
lim
x→0
lim
x
ln(1 − 7x)
x→+∞
lim
x→0
(3x5 − 2x3 + x + 7)4
(x2 + 4)10
x
arcsin(x/3)
lim+ x ln(x)
5
x→0
x→0
2.


 x + 2 , −3 < x < −2 ,
−x − 2 , −2 ≤ x < 0 ,
Is the function f , defined by f (x) =

 x + 2 , 0 ≤ x < 1,
continuous or not?
3.
Determine lim+ sin arctan(ln(x)) .
x→0
4.
Explain why the series in (a)–(b) are geometric (the symbol ”...” here denotes
all other terms of a geometric series). Then find which of them are convergent.
Also, find the sums of those series which are convergent.
a)
9 + 3π + π 2 + . . .
b)
∞
P
n=2
5.
Find the power series in x, i.e.
∞
P
1 n
2
c)
3−
9 27
+
− ...
4 16
cn xn , which represents the expression x/(1 −
n=0
3x). In particular, specify the interval of convergence of the series and the
coefficients of the series.
MAA151 Single Variable Calculus academic year 2016/17
Exercises in class A – Set 3 (5) – Differentiation, applications of differentiation
1.
2.
Find the expressions for the derivatives and write the answers in as simple form
as possible.
a)
d 3
x ln(x)
dx
d)
d3
cosh(x)
dx3
g)
d4
sin(x)
dx4
b)
d 2x + 3
dx x2 + 2
e)
d
(5x + 1)17
dx
h)
d 3x2 −7x
e
dx
c)
√
d
arctan( x − 1)
dx
f)
d
ln |1 + cos(x2 )|
dx
i)
d
arcsin(3x)
dx
The function f is differentiable, and it is known that
f (0) = 2 , f (1) = −2 , f (2) = 5 , f (3) = 4 ,
f 0 (0) = 1 , f 0 (1) = 3 , f 0 (2) = 7 , f 0 (3) = −2 .
Find an equation for the tangent line to the curve y = f (2x2 − 3x + 1) at the
point P whose x-coordinate is equal to 2 .
3.
Prove that the function x y f (x) = x3 + 2x is invertible. Then, find the
derivative of the inverse function f −1 at the point 12.
4.
Classify all the local extreme points and find the range of the function f definied
by
f (x) = x4 − 2x2 − 8 , Df = [−2, 3] .
Then roughly sketch the curve y = f (x) and find the intervals where the function
is convex and concave respectively. Also, state the inflection points of its curve.
5.
The weighted sum of two non-negative numbers is 2. Which are the numbers if
the weights of the first and the other are 3 and 4 respectively, and the sum of
the first and the cube of the other is a minimum? Prove your conclusion.
6.
Let γ :
x = t3/2√,
y = 1/ t .
Find an equation for the tangent line τ to the curve γ at the point P : (8, 21 ).
7.
√
Which point on the curve γ : y = x x is closest to the point P : (1, 0)?
MAA151 Single Variable Calculus academic year 2016/17
Exercises in class A – Set 4 (5) – Antiderivatives, differential equations
1.
Find the general antiderivatives of . . .
Z
x
5x dx
√
a) x y
b)
x+1
(x + 3)(x − 2)
c)
x y (2x − 3) cos(x)
2.
Solve the initial value problem xy 0 + 2y = x , y(1) = 2 .
3.
Find the general solution to the differential equation 6y 00 + 5y 0 + y = 0 .
4.
Find all antiderivatives of . . .
a)
xy
x
1 + (x + 1)2
b)
x
xy p
1 − (x + 1)2
(
c)
xy
x2 + 1
5x3 + 15x
y 00 + 2y 0 + y = 0 ,
5.
Solve the initial value problem
6.
Find to the differential equation y 00 + 4y = 0 the solution which satisfies the
initial conditions y( π2 ) = y 0 ( π2 ) = −2.
y(0) = −5 , y 0 (0) = 12 .
MAA151 Single Variable Calculus academic year 2016/17
Exercises in class A – Set 5 (5) – Integrals
1.
Find the area of the region which in the first quadrant is enclosed by the curves
y = x och y = 2x/(1 + x2 ) .
2.
Use the Taylor expansion of order 2 for the√function x y
to find the best possible approximation to 3 9.
f
Z
3.
√
3
x about the point 8
3/2
| ln(x)| dx .
Evaluate the integral
1/2
4.
Find for each of the functions f1 , f2 , f3 , f4 , definied by

f1 (x) = cosh(x) ,





 f2 (x) = |x| + 5/x ,
f3 (x) = x2 sin(x) + x/ cos(x) ,




√

 f4 (x) = x ,
whether it is even, odd, or neither even nor odd.
Z
5.
3
Z
5
Prove that the integral
x sin(x) dx can be expressed as 2
−3
Z
6.
5
Evaluate the integral
0
2
x59 ex sin4 (x) dx .
−5
7.
Evaluate the integral
Z
√
13/ 2
√
169 − x2 − x dx
0
by interpreting it as the area of a certain region.
3
x5 sin(x) dx .