Download MA10192 Mathematics 1

MA10192 Mathematics 1
Mark Opmeer
Additional Exercises
Chapter 1
Functions and equations
1.1
Polynomials
Solve the following equations for x:
1. x2 − 2x + 1 = 0,
2. x2 + 2x + 2 = 0.
1.2
Exponentials and logarithms
Solve the following equations for x (remember |y| equals y if y ≥ 0 and −y if
y < 0):
1. ex e2x = 1,
2. e2x − 2ex = 1,
3. ex e3x = 2,
4. ex = 2e−x ,
5. e2x − 2ex + 1 = 0,
6. ln x = 3.
7. ln |x − 2| + ln |x − 3| = 0,
8. ln (x + 1) + ln (x − 4) = 0,
9. ln |x2 + 2x − 2| = 0,
10. ln (x − 1) + ln (x + 2) = 0,
1
1.3
Trigonometric functions
1.3.1
Definitions
1.3.2
Solving simple trigonometric equations
Solve the following equations for x:
√
1. cos x =
3
2 ,
2. sin 2x =
√1 ,
2
3. tan 2x = 0,
√
3
2 ,
4. sin x =
5. cos 2x =
6. sin x =
√1 ,
2
√1 ,
2
7. cos x sin x = 0.
1.3.3
Inverse trigonometric functions
1. Solve sin x =
1.3.4
1
6
for x.
Polar coordinates
1. Write the point given in Cartesian coordinates by (1, −2) in polar coordinates.
2. Write the point given in polar coordinates as r = 2, θ =
coordinates.
1.3.5
3π
4
in Cartesian
Harmonic form
1. Solve 2 sin x + 4 cos x = 3 for x.
2. Solve sin x + 5 cos x = 3 for x.
1.3.6
Solving more complicated trigonometric equations
1. Solve sin2 x + cos2 2x = 1 for x.
2. Solve cos2 x + sin2 2x = 1 for x.
2
Chapter 2
Differentiation
2.1
Definition
1. Obtain the derivative of y = 2x3 from first principles (meaning: look at
the slope of chords etc.).
2. Obtain the derivative of y = x4 from first principles (meaning: look at the
slope of chords etc.).
3. Obtain the derivative of y = x3 + x2 from first principles (meaning: look
at the slope of chords etc.).
4. Find the tangent line of y = 3x5 in the point (1, 3) (you may use the rules
for differentiation).
5. Find the tangent line of y = x5 + x2 + 1 in the point (1, 3) (you may use
the rules for differentiation).
2.2
Notation
2.3
Derivatives of standard functions
2.4
Rules for differentiation
1. Calculate the derivative of x2 cos x.
2. Calculate the derivative of tan2 3x.
3. Calculate the derivative of
1
x2 .
4. Find the derivative of tan x + xe3x .
5. Calculate the derivative of x2 ex .
3
6. Calculate the derivative of
tan x
x2 +x+2 .
7. Compute the derivative of x sin x.
8. Compute the derivative of cos(ex ).
9. Compute the derivative of
2.5
ln x
x .
Derivatives of inverse functions
1. Calculate the derivative of arccos2 3x.
2.6
Parametric differentiation
dy
at the point (1, 0) of the curve given by the equa1. Calculate the slope dx
t
2
tions x(t) = e + t , y(t) = tan t.
dy
2. Calculate the slope dx
at the point (1, −1) of the curve given by the
equations x(t) = ln t + t4 , y(t) = sin πt + cos πt.
dy
at the point (x, y) = (0, 1) for the curve given paramet3. Find the slope dx
rically by x(t) = t sin(t), y(t) = et cos t.
2.7
Implicit differentiation
dy
1. Calculate the slope dx
at the point (0, 0) of the curve given by the equation
x
sin y + e cos 3y = 1.
dy
2. Calculate the slope dx
at the point (0, 0) of the curve given by the equation
y
y sin x + e cos y = 1.
3. Compute the derivative
dy
dx
for the function given implicitly by
ey + ln(xy) + x2 = 0.
2.8
Maxima and minima
1. Determine the location of the local maxima and minima of the function
x3 + 3x2 − 9x + 5 for −4 ≤ x ≤ 4. Be sure to also determine whether the
points found are a local maximum or a local minimum.
2. Determine when the function 2x3 − 15x2 + 36x + 2 is increasing and when
it is decreasing.
3. Determine the location of the local maxima and minima of the function
2x3 + 3x2 − 12x − 5 for −4 ≤ x ≤ 4. Be sure to also determine whether
the points found are a local maximum or a local minimum.
4
4. Determine when the function x3 − 4x2 + 2x + 3 is increasing and when it
is decreasing.
5. Determine the location of the local maxima and minima of the function
ex (x2 − 2x + 1) for −2 ≤ x ≤ 2. Be sure to also determine whether the
points found are a local maximum or a local minimum.
2.9
Asymptotes
1. Determine the horizontal and vertical asymptotes of the function
x+3
x2 −4 .
2. Determine the horizontal and vertical asymptotes of the function
x2 +3
x2 −9 .
3. Determine the horizontal and vertical asymptotes of the function
sin x
ex −e−x .
4. Determine the horizontal and vertical asymptotes of the function
2x
x2 −1 .
5. Determine the horizontal and vertical asymptotes of the function
2.10
Numerical solution of equations
2.11
Taylor polynomials
x arctan x
.
x−1
1. Calculate the Taylor polynomial of degree 8 of x sin (x2 ) around x = 0.
2. Calculate the Taylor polynomial of degree 10 of x cos (x3 ) around x = 0.
3. Determine the Taylor polynomial of degree 3 of e2x (x2 + 2x + 1) around
x = 0.
4. Determine the Taylor polynomial of degree 5 of ln x around x = 1.
5
Chapter 3
Integration
3.1
Antiderivatives
1. Compute
R
sin x dx.
2. Compute
R
ex dx.
3.2
Estimating area
1. Compute the area between the lines y = x, x = 0, x = 1 and y = 0
by approximating it from above and below by the area of rectangles and
taking the limit.
2. Compute the area between the lines y = x3 , x = 0, x = 1 and y = 0
by approximating it from above and below by the area of rectangles and
taking the limit.
3. Compute the area between the curves y = x2 , x = 0, x = 2 and y = 0
by approximating it from above and below by the area of rectangles and
taking the limit.
3.3
Definition
3.4
The fundamental theorem of calculus
1. Compute
R π/2
2. Compute
R1
3. Compute
R1
0
0
0
cos x dx.
x2 dx.
ex dx.
6
3.5
3.5.1
Rules for integration
The substitution rule
e2x dx.
1. Compute
R
2. Compute
R1
3. Compute
R
4. Compute
R1
5. Compute
R
sin 3x dx.
6. Compute
R
x cos(x2 ) dx.
3.5.2
x
0 x2 +1
dx.
e3x dx.
x
0 x2 +1
dx.
Integration by parts
1. Compute
R
x sin x dx.
2. Compute
R
ex sin x dx.
3. Compute
R
x2 cos x dx.
4. Compute
R
x3 ex dx.
3.6
Anti-derivatives of inverse functions
1. Compute
3.7
R
arctan x dx.
Anti-derivatives of rational functions by partial fractions
1. Compute
R
x−2
(x+1)2 (x−1)
2. Compute
R
1
x2 +4
dx.
3. Compute
R
1
x2 −4
dx.
4. Compute
R
2x−4
(x2 +1)2
5. Compute
R
x+1
x2 −3x+2
6. Compute
R
x+1
x2 +4
dx.
7. Compute
R
1
x2 −1
dx.
8. Compute
R
1
x2 +2
dx.
dx.
dx,
dx.
7
3.8
Anti-derivatives of trigonometric rational functions
1. Compute
R
cos x sin5 x dx.
2. Compute
R
cos3 x
sin x
3. Compute
R
cos2 x dx.
4. Compute
R
cos3 x sin3 x dx.
3.9
dx.
Trigonometric substitution
1. Calculate
R√
3 − x2 dx.
2. Calculate (1 − x2 )3/2 dx.
R
3.10
Hyperbolic substitution
1. Compute
3.11
R√
1 + x2 dx.
Area between curves
1. Calculate the area bounded by the lines x = 2, x = 5 and the graphs of
y = x2 and y = 4x.
2. Calculate the area in the quadrant x ≥ 0, y ≥ 0 bounded by the the
graphs of y = 2x3 and y = 3x.
3. Calculate the area in the quadrant x ≥ 0, y ≥ 0 that is bounded by the
curves y = x2 and y = 2x.
3.12
Numerical integration
3.13
Improper integrals
1. Calculate
R1
2. Calculate
R∞
3. Calculate
R1
4. Compute
R1
0
1
√1
x
dx.
1
√
x x
1
0 x
dx.
dx.
ln x dx.
0
R∞ 1
5. Compute 1 x3 dx.
8
Chapter 4
Ordinary differential
equations
4.1
Separable equations
1. Solve
dy
dx
= yx.
2. Solve
dy
dx
=
4.2
x
y
with initial condition y(0) = −1.
Linear equations
1. Solve
dy
dx
− y = e2x .
2. Solve
dy
dx
+ y tan x = sin 2x with initial condition y(0) = 1.
3. Solve
dy
dx
+ 5y = x, y(0) = 2.
4.3
Numerical methods
9
Chapter 5
Functions of several
variables
5.1
Partial derivatives and extrema
1. Compute the partial derivatives
sin(xy) + y 2 + x.
∂f
∂x
and
∂f
∂y
for the function f (x, y) =
2. Find the critical points (i.e. the points where ∂f
∂x = 0,
2
2
2
the function f (x, y, z) = x + y + 3z + x − 2y + 5.
10
∂f
∂y
and
∂f
∂z
= 0) for