# Math101, 2019 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. The activity consists of 9 sessions for all participants. In Aalborg, Math101 is available in Danish, whereas it is 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.

### Enrolment

To enrol in the course, please write an email to math101_tilmelding@staff.aau.dk.

### Supplementary material

Below, you will find supplementary material grouped by topic.

#### Webmatematik (Danish)

Defining derivatives calculus-derivatives-1-new-hd-version
derivative-intuition-module
derivative_intuition differenskvotient-og-differentialkvotient
Derivatives power-rule
derivative-properties-and-polynomial-derivatives
derivatives-of-sin-x-cos-x-tan-x-e-x-and-ln-x
power_rule
special_derivatives
afledede-funktioner
regneregler-for-differentialkvotienter
differentiation-af-trigonometriske-funktioner
Derivatives of composite functions chain-rule-introduction
chain-rule-definition-and-example
chain-rule-for-derivative-of-2-x
chain-rule-with-triple-composition
chain_rule_1
chain-rule-on-three-functions
differentiation-af-sammensat-funktion
Product and quotient rules applying-the-product-rule-for-derivatives
product-rule-for-more-than-two-functions
quotient-rule-from-product-rule
product_rule
quotient_rule
Area calculations area-between-a-curve-and-an-axis bestemt-integral-og-areal
areal-mellem-to-grafer
Optimization minimizing-sum-of-squares
minimizing-the-cost-of-a-storage-container
optimization optimering
Integration and antiderivatives antiderivatives-and-indefinite-integrals
indefinite-integrals-of-x-raised-to-a-power
antiderivative-of-hairier-expression
basic-trig-and-exponential-antiderivatives
antiderivatives
indefinite-integrals
Integration by substitution u-substitution
u-substitution-and-back-substitution
integration-by-the-reverse-chain-rule
integration-by-u-substitution
integration-ved-substitution

### Supplementary exercises

Below, you will find supplementary exercises grouped by topic.

#### Powers, roots, and fractions

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.

#### Rewriting expressions

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.

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.

#### Functions

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.

#### Derivatives and integrals (A-level)

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.

#### Derivatives and integrals (B-level)

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.

#### Differential equations

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.

#### Indefinite integrals

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.