# Math101, 2018 autumn

## Introduction

Have you always been struggling to understand mathematics or have you never got the necessary routine to carry out basic mathematical manipulations? This is your opportunity to change that. Math101 is an activity you should consider, if you are determined to improve your basic mathematical skills that are required by the mathematical courses on your first year of study.

Maybe you are in doubt whether the activity is aimed at you. To find out, you can take this test, which you should be able to complete without the use of any aids and only with a few or no mistakes. The answers can be found here. If you had more than a couple of mistakes, or used more than two hours to complete the test, and are you determined to work with your basic mathematical skills, then you should consider following the Math101 activity. If you need another test, you can find one here. It is a bit more difficult and without answers.

Math101 is planed so that students with both A and B level from high school can attend. In Aalborg, the activity consists of 9 sessions for all participants. In Esbjerg and Copenhagen, the activity consists of 7 sessions, where 5 are shared, and the last 2 are split in A and B level sessions. In Aalborg, Math101 is available in Danish and English, whereas it is only available in English in Esbjerg and Copenhagen. Part of the course material in Aalborg, however, will also be in English.

### Course material

In Aalborg, the course material is found at the corresponding Moodle page, while the activities in Esbjerg and Copenhagen follow the plan on these pages.

### Criteria to pass

This course is to be passed by active participation. Active participation means that you max may be absent from one of the seven lectures.

### Course plan

Lecture Topic Content Objective
1 Calculations with numbers Introduction to numbersets - specially the rational and irrational numbers. Orders of operation, exponents, radicals and fractions.
• The students must learn to look at fractions and irrational numbers as numbers and not only calculations.
• The students must learn to give an exact result that is as reduced as possible
• It must not be a surprice to the student that $$\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$$
2 Reduction The students mus learn to reduce expressions with one or more unknown. Also recuctions that include taking the square of a binomial or factorization is included.
• In this lecture it shold be time to focus on how fast the students carry out the manipulations
3 Solving equations Solving linear and quadratic equations.
• The students must easily solve linear and quadratic equations with integer coefficients.
• The students must be able to solve linear and quadratic equations with rational coefficients.
• The students must be able to solve linear equations with irrational coefficients.
4 Functions Functions and inverse functions
• The students must know the function $f(x)=x^2$, the trigonometric functions and the exponential functions.
• The students must know how to use the inverse functions to solve simple equations
• The students must know about exact values for the trigonomentric functions for simple angles.
5A Differential- og integral calculus Rules for differentiation og integration
• The students must easily be able to calculate derivatives and integrals of sums, products and quotients of simple functions.
• The students must be able to calculate derivatives of "simple" combined functions.
• The students must be able to performe integration by substitution of "simple" functions.
5B Differential calculus Rules for finding derivatives
• The students must be familiar with the concept of derivatives.
• The students must easily be able to find derivatives of sums of "simple" functions.
• The students must be able to solve "simple" optimization problems.
6A Differential equations Solution of linear differential equations
• The students must be able to easily solve "simple" linear differential equations, such as $y' = ky$.
• The students must also be able to solve more general linear equations.
6B Integral calculus Rules for integration
• The students mus be able to determine both definite and indefinite integrals of "simple" functions
• The students must be able to interpretate definite integrals as the area between functions
• When doing indefinite integrals the students must be able to determine the arbitry constant af integration from a given condition.
7 Problem solving Problem solving
• The students are doing a set of problems.

### Materials

Press "Lecture 1" to see the materials for lecture 1, and similar for the other lectures.

### Exercises

Press "Lecture 1" to see the exercises for lecture 1, and similar for the other lectures.

#### Lecture 1

Exercise 1. Calculate the following numbers:
1. $3 \cdot 2 - \frac{15}{3}+\frac{8}{4}-4$
2. $-6^2$
3. $(-6)^2$
4. $3 \cdot (5-7)-2+3 \cdot 4-3 \cdot (6-9)$
Exercise 2. Calculate the following numbers:
1. $\frac{3}{4} + \frac{5}{3}$
2. $\frac{3}{8} \cdot \frac{2}{8}$
3. $\frac{3}{7} \cdot \frac{2}{3}$
4. $\frac{3}{4}-\frac{2}{3} \cdot \frac{7}{4}$
5. $\frac{2}{3} : \frac{4}{9}$
6. $6 \cdot \frac{2}{5}$
7. $6 : \frac{2}{5}$
8. $\frac{3}{4} \cdot \frac{5}{4}-\frac{1}{2} \cdot \frac{3}{8}$
9. $\frac{1}{4} +2 \cdot \frac{4}{16}$
Exercise 3. Calculate the following numbers:
1. $(\sqrt{4})^2$
2. $(\sqrt{2})^2$
3. $(-\sqrt{2})^2$
4. $(\sqrt{2})(1+\sqrt{2})$
5. $3\sqrt{8}+\sqrt{2}(3-2\sqrt{2})$
6. $\frac{2\sqrt{14}+4\sqrt{63}}{\sqrt{2((2\sqrt{2})^2+10}}$
Exercise 4. Rewrite in simplest form with exponents using basenumbers 2 and 3:
1. $3^2 \cdot 3^5 \cdot 3^{-3}$
2. $\frac{3^2 \cdot 3^5 \cdot 3^{-4}}{3^3 \cdot 3^{-6}}$
3. $\frac{(\frac{3}{4})^3 \cdot 2^4 \cdot (3^{-2})^3}{3^{-3} \cdot 2^{10}}$
4. $3^4 \cdot 6^2 \cdot 12^3 \cdot 6^4$
5. $\frac{2^3 \cdot 6^3 \cdot 12^3 \cdot 3^{-8}}{4^2 \cdot 9^3}$

Exercise 5. Get together with another student that has finished the above exercises. Make $2 \times 5$ exercises of the same type as above and solve the exercises made by the other student. You have to be able to solve your own exercises, so you can give a hint to the other student if he/she cannot solve your exercises.

#### Lecture 2

Exercise 1. Reduce the following expressions:
1. $3x+5y-2xy+7x-2xy+2y$
2. $3x^2+3x+4y-3x+2y+y^2$
3. $2x^2+3xy-yx+x^2$
4. $3x(x-1)-2(1+x^2)$
5. $(2+x)(x-1)-4(x-1)$
6. $x(2x-1)-(3x-2)(x-1)$
7. $(x-1)(1-\frac{1}{x})+2(1-\frac{x}{2})$
Exercise 2. Reduce the following expressions:
1. $(x+3)^2$
2. $(4x-5)^2$
3. $(a-2)^2-a(a-1)$
4. $(s-1)(s+1)+1$
5. $(x+\sqrt{3})^2-\sqrt{3}(x+2\sqrt{3})$
6. $(x+\sqrt{6})(x-\sqrt{6})-\sqrt{6}(x-\sqrt{24})$
Exercise 3. Reduce the following fractions:
1. $\frac{9a^2b^5}{3(ab)^3}$
2. $\frac{6a^3b^{-4}}{(2a^2b)^2}$
3. $\frac{2x^{-4}y^3}{(2y^2x)^{-2}}$
4. $\frac{(x+3)^2}{2x^2+6x}$
5. $\frac{4x^2-9}{4x^2+9-12x}$
6. $\frac{2x^2+18+12x}{x^2+3x}$
Exercise 4. Determine whether the following expressions are true or false:
1. $\frac{1}{1+x}=\frac{2}{2+x}$
2. $\frac{\frac{1}{b}+1}{1-\frac{a}{b}}=\frac{a+b}{b-a}$
3. $\frac{\frac{a}{b}+1}{\frac{b}{a}+1}=\frac{a}{b}$
4. $\frac{1}{1+x}=1+\frac{1}{x}$

Exercise 5. Get together with another student that has finished the above exercises. Make $2 \times 5$ exercises of the same type as above and solve the exercises made by the other student. You have to be able to solve your own exercises, so you can give a hint to the other student if he/she cannot solve your exercises.

#### Lecture 3

Exercise 1. Solve the equations:
1. $8x+2=26$
2. $-3x-5=4$
3. $-6x+7=-29$
4. $8x+11=5$
5. $-7x+4=-8$
Exercise 2. Solve the equations:
1. $3x+7=-2x+2$
2. $-3x-4=-x+3$
3. $3(x-4)+2=2(x+1)$
4. $-2(x-1)+2=3x+2(x-7)$
Exercise 3. Solve the equations:
1. $3(x-2)+2=3x-8$
2. $-(x+1)+2x=2(x-1)-(x-1)$
3. $4(x+1)=2x(\frac{1}{x}+1)$
Exercise 4. Løs ligningerne:
1. $\frac{2}{3}(x-\frac{5}{2})=\frac{3}{6}$
2. $\frac{4}{3}(\frac{3}{5}x-\frac{2}{3})=-\frac{3}{2}(x-\frac{2}{5})$
3. $\frac{3}{8}(4x-2)=-\frac{1}{3}(x-\frac{3}{4})$
4. $\frac{1}{7}(\frac{2}{3}x-\frac{3}{3})-\frac{3}{7}x=-\frac{1}{2}(x-5)$
5. $\sqrt{2}x+4=8$
6. $\pi (x-1)=\sqrt{2}x+3$
7. $\sqrt{2}(2\sqrt{2}x-\sqrt{8})=2x+1$
Exercise 5. Solve the equations:
1. $x^2=36$
2. $x^2=-81$
3. $(x-1)(x+2)=0$
4. $x^2+x-1=0$
5. $2x^2+3x+1=0$
6. $x^2+4x+3=0$
Exercise 6. Solve the equations:
1. $x^2-3x-1=-2$
2. $-2x^2+3x+1=0$
3. $-3x^2+3x-2=8$
4. $-x^2+4x-2=2$
5. $-2x^2+3x-4=2(x-1)$

Exercise 7. Get together with another student that has finished the above exercises. Make $2 \times 5$ exercises of the same type as above and solve the exercises made by the other student. You have to be able to solve your own exercises, so you can give a hint to the other student if he/she cannot solve your exercises.

#### Lecture 4

Exercise 1. Calculate the numbers:
1. $2^5, \quad3^4, \quad 3^3, \quad 4^2$
2. $\log_2 2^5,\quad \log_3 3^4,\quad \log_3 3^4,\quad \log_4 4^{-2}$
3. $\log_2 (256),\quad \log_{10}(1000),\quad \log_2 (\frac{1}{8}),\quad \log_{10}(0.00001)$
Exercise 2. Calculate the numbers:
1. $\log (40)+\log(25)$
2. $\log(40)-\log(4)$
3. $\log(2)+\log(30)-\log(6)$
4. $10^{\log(7)},\quad 10^{1+\log(3)}, \quad 10^{2\log{4}}, \quad 10^{-\log{7}}$
5. $\ln(e^{5}), \quad e^{\ln(9)},\quad \ln(e^3)-\ln(e)$
Exercise 3. Solve the equations:
1. $\ln (x)=4$
2. $e^x=5$
3. $3 \log(x)=\log(27)$
4. $\ln(2x-4)=\ln(8)+\ln(4)$
Exercise 4. Calculate:
1. $\sin (\frac{\pi}{4}) + \cos (\frac{\pi}{4})$
2. $\tan(\frac{\pi}{6}) + \cos(\frac{\pi}{3})$
3. $\frac{\sin (\frac{\pi}{6}) + \cos (\frac{\pi}{3})}{\sin(\frac{4\pi}{6})}$
4. $\frac{\cos (\frac{\pi}{6}) + \sin (\frac{\pi}{2})}{\tan(-\frac{\pi}{3})}$
Exercise 5. Find one solution for the equations:
1. $\sin(x)=\frac{\sqrt{3}}{2}$
2. $\cos(x-\pi)=-\frac{\sqrt{2}}{2}$
3. $2 \sin^2(x)+5\sin(x)+2=0,$ substitute $u=\sin(x),$ and solve for $u$

Exercise 6. Get together with another student that has finished the above exercises. Make $2 \times 5$ exercises of the same type as above and solve the exercises made by the other student. You have to be able to solve your own exercises, so you can give a hint to the other student if he/she cannot solve your exercises.

#### Lecture 5A

Exercise 1. Find the derivative of the functions:
1. $f(x)=3x^7+2x^4-3x^2$
2. $f(x)=2x^5+3x^{\frac{3}{2}}-2x^{-2}$
3. $3 \sqrt{x}+\frac{1}{x}$
4. $3e^{2x}+3^x$
Exercise 2. Use the product and quotient rule to determine the derivative $f'(x):$
1. $f(x)=3xe^x$
2. $f(x)=2x^2\sin(x)$
3. $f(x)=\frac{3x^2+2x-1}{x-1}$
4. $f(x)=\frac{2x^5-2x^3+1}{x^4-2x}$
Exercise 3. Find the derivative of the composite functions - Do not reduce the expression:
1. $f(x)=\sqrt{x^2+2x-1}$
2. $f(x)=(x^3+2x)^{\frac{3}{2}}$
3. $f(x)=\sin(x^2+1)$
4. $f(x)=\cos ^2 (3x^3+x-1)$
5. $f(x)=\cos ^2 ((x-1)^5)$
Exercise 4. Determine the indefinite integrals:
1. $\int x^3+2x^2-1 dx$
2. $\int \sin(x)+e^x dx$
3. $\int \ln(x)+e^x dx$
Exercise 5. Use integration by substitution to determine the indefinite integrals:
1. $\int 2x \sin (x^2-1)$
2. $\int (3x^2+2) \cos(x^3+2x+7) dx$
3. $\int x e^{3x+1}dx$
4. $\int (x+1)\sin (x-3)dx$

Exercise 6. Get together with another student that has finished the above exercises. Make $2 \times 5$ exercises of the same type as above and solve the exercises made by the other student. You have to be able to solve your own exercises, so you can give a hint to the other student if he/she cannot solve your exercises.

#### Lecture 5B

Exercise 1. Find the derivative of the followin functions:
1. $f(x)=3x^7+2x^4-3x^2$
2. $f(x)=2x^5+3x^{\frac{3}{2}}-2x^{-2}$
3. $3 \sqrt{x}+\frac{1}{x}$
4. $3e^{2x}+3^x$
5. $\frac{x^2}{3}-5\sqrt{x}$
6. $e^{3x}+\frac{1}{3}\ln(x)$
7. $7\sqrt{x}-\ln(x)+\frac{1}{x^2}$
Exercise 2.Determine the antiderivative, $F(x)$, for the following functions with the given condition:
1. $f(x)=x+5$, and $F(2)=4$
2. $f(x)=-\frac{2}{x}, \quad x>0$, and $F(e)=1$
3. $f(x)=4e^{2x}$, and $F(\frac{1}{2})=e$
Exercise 3. Calculate the indefinite integrals:
1. $\int x^3+2x^2-1 dx$
2. $\int \frac{2}{x}+e^x dx$
3. $\int \ln(x)+e^x dx$
4. $\int \frac{2}{6\sqrt{x}}+5x^{-3}dx$

Exercise 4. A $300$ m long fence must enclose a rectangular area. Determine the lenght of the sides in the rectangle, so that the area is as large as possible.

Exercise 5. A $300$ m long fence must enclose a rectangular area. One side is along a river, so the fence is only needed on three sides. Determine the lenght of the sides in the rectangle, so that the area is as large as possible.

Exercise 6.An open box must have a quadratic bottom and a volume of $5000 cm^3$. Determine the side length of the bottom of the box and the height, so that the surface area is as small as possible.

Exercise 7.A rectangle is inscribed into a halfcircle with radius 12 m. Determine the length and width of the rectangle, so its area is as large as possible.

Exercise 8. Get together with another student that has finished the above exercises. Make $2 \times 5$ exercises of the same type as above and solve the exercises made by the other student. You have to be able to solve your own exercises, so you can give a hint to the other student if he/she cannot solve your exercises.

#### Lecture 6A

Exercise 1. Determine whether or not the function $f$ is a solution to the given differential equation
1. $f(x)=\frac{1}{3}x^3-6$ and $y'=x^2$
2. $f(x)=x-1$ and $y'=\frac{y+1}{x}$
3. $f(x)=2 e^{x}$ and $\frac{y'}{y^2}=\frac{1}{2} e^{-x}$
4. $f(x)=x+\frac{1}{x}$ and $xy'=2x-y$
5. $f(x)=4 e^{3x}-e^{2x}$ and $y'-2y=e^{2x}$
6. $f(x)=\ln (e^{x} + e - 1)$ and $y'=e^{x-y}$
Exercise 2. The given differential eguation has a solutioon with a graph through the given point. Write an equation for the tangent line through that point
1. $y'=\frac{x+2y}{3}$, $(1,3)$
2. $y'=x-y^2$, $(3,5)$
3. $y'=(2x+1)y^2$, $(0,-1)$
4. $y'=3\sqrt{x}\sqrt{y}$, $(4,1)$
Exercise 3. Find the solution with a graph through the given point for the differential equations
1. $y'=6y$, and $(\ln (2),100)$
2. $y'-y=0$, and $(\ln (4),28)$
3. $y'=-3y$, and $(3,8)$
4. $y'=0,02y$, and $(0,41000)$
5. $y'=32-8y$, and $(3,10)$
6. $y'=-0,1y+25,4$, and $(0,180)$
Exercise 4. Find the solutions for the given differential equations
1. $y'+3x^2y=6x^2$
2. $y'+2xy=0$, $(1,e^{-1})$
3. $xy'+y=\frac{1}{x}$, $(1,4)$

Exercise 5. Find the constants $a$ and $b$ so the function $f(x)=ax+b$ is a solution for the differential equation $y'-y=2x-3$.

Exercise 6. Get together with another student that has finished the above exercises. Make $2 \times 5$ exercises of the same type as above and solve the exercises made by the other student. You have to be able to solve your own exercises, so you can give a hint to the other student if he/she cannot solve your exercises.

#### Lecture 6B

Exercise 1: Is the function $F(x)$ an antiderivative of $f(x)$
1. $F(x)=x\ln (x) - x$, $f(x)=\ln (x)$
2. $F(x)=2x\sqrt{x} - x$, $f(x)=3\sqrt{x}-1$
Exercise 2: Find an antiderivative
1. $f(x)=4x+5$
2. $f(x)=x^2+2x-9$
3. $f(x)=0$
4. $f(x)=2x-\frac{3}{\sqrt{x}}$
Exercise 3: Find the following indefinite integrals
1. $\int 4x^3 + \frac{8}{x^2} dx$
2. $\int 7 e^x-e^{7x} dx$
3. $\int e^{2x}+\frac{4}{5\sqrt{x}} dx$
Exercise 4: Find the antiderivative $F(x)$ that satisfies the condition
1. $f(x)=3x^3+2x^{-2} \mbox{ , } F(1)=1$
2. $f(x)=-\frac{7}{x}+\sqrt{x} \mbox{ , } F(0)=4$
3. $f(x)=e^x+2e^{-2x} \mbox{ , } F(\ln (2))=1$
Exercise 5: Calculate the definite integrals
1. $\int _{-3}^{1} x^2 -7x +1 dx$
2. $\int _{1}^{4} x^{\frac{1}{2}}+3x^{-\frac{1}{2}} dx$
3. $\int _{0}^{1} e^x dx$

Exercise 6: Calculate the area bounded by the graph of the function $f(x)=9-x^2$ and the first coordinate axis.

Exercise 7: Calculate the area bounded by the graphs of the functions $$f(x)=\frac{1}{4}x^2 - 2\quad \mbox{ and } \quad g(x)=x+1.$$

Exercise 8: Get together with another student that has finished the above exercises. Make $2 \times 5$ exercises of the same type as above and solve the exercises made by the other student. You have to be able to solve your own exercises, so you can give a hint to the other student if he/she cannot solve your exercises.