BasalMat (Basic Math)

Topic	Khan Academy	Khan Academy exercises	Webmatematik (Danish)
Numbersets	Numbersets 1 Numbersets 2 Introduction to rational and irrational numbers		Tal
Order of operations	Order of Operations 1	order of operations 2	Regnearternes-hierarki
Exponents	Introduction to negative exponents products and exponents raised to an exponent properties capstone-exponent-properties-example	Exponents 2 properties of integer exponents Exponent rules	Potenser
Radicals	simplifying radicals more-simplifying-radical-expressions	simplifying_radicals	Kvadratrødder og andre rødder
Fractions	adding-fractions-with-unlike-denominators subtracting-fractions-with-unlike-denominators multiplying-fractions-and-whole-numbers multiplying-fractions-and-whole-numbers multiplication-as-scaling another-dividing-fractions-example examples-of-dividing-negative-fractions	adding_and_subtracting_radicals adding fractions subtracting fractions adding and subtracting fractions multiplying fractions by integers Fraction multiplication as scaling examples of dividing negative fractions	addition-og-subtraktion-af-brøker multiplikation-og-division-med-brøker

Topic	Khan Academy	Khan Academy exercises	Webmatematik (Danish)
Combining like terms	combining-like-terms 2 combining-like-terms-3 combining-like-terms-and-the-distributive-property	combining_like_terms_1 combining_like_terms_2
Parentheses	multiplying-and-dividing-monomials-3 monomial-greatest-common-factor multiplication-of-polynomials	manipulating-linear-expressions-with-rational-coefficients manipulating-linear-expressions-with-rational-coefficients nested-fractions
Square	square-a-binomial special-products-of-binomials	multiplying_expressions_0.5 multiplying_expressions_1
Factorization	factoring-and-the-distributive-property-3 factoring-polynomials-1 factor-by-grouping-and-factoring-completely	factoring_polynomials_1 factoring_polynomials_2 factoring_polynomials_by_grouping_1

Topic	Khan Academy	Khan Academy exercises	Webmatematik (Danish)
Linear equations	adding-and-subtracting-the-same-thing-from-both-sides intuition-why-we-divide-both-sides why-we-do-the-same-thing-to-both-sides-multi-step-equations two-step-equations solving-equations-2 number-of-solutions-to-linear-equations-ex-3	linear_equations_2 linear_equations_3 understanding-the-process-for-solving-linear-equations solutions-to-linear-equations solving_for_a_variable	ligninger
Two equations with two unknown	systems-with-elimination-practice	systems_of_equations_with_elimination_0.5 systems_of_equations_with_elimination	to-ligninger-med-to-ubekendte grafisk-losning-af-to-ligninger-med-to-ubekendte
Quadratic equations	using-the-quadratic-formula quadratic-formula-3	quadratic_equation	andengradsligningen

Topic	Khan Academy	Khan Academy exercises	Webmatematik (Danish)
Functions	what-is-a-function understanding-function-notation-example-2	functions_1	hvad-er-en-funktion
Inverse functions	introduction-to-function-inverses function-invertibility making-function-invertible solving-inverse-function-equation	inverses_of_functions algebraically-finding-inverses understanding-inverse-functions	omvendte-funktioner
Exponential and logarithms	exponential-growth-functions logarithms comparing-exponential-logarithmic-functions introduction-to-logarithm-properties introduction-to-logarithm-properties-part-2 logarithm-of-a-power	logarithms_1 using-logarithms-to-solve-exponential-equations logarithms_2	eksponentiel-udvikling logaritmer
Trigonometric functions	unit-circle-definition-of-trig-functions-1 trig-functions-special-angles inverse-trig-functions-arcsin inverse-trig-functions-arccos inverse-trig-functions-arctan	unit_circle trigonometric-functions-of-special-angles inverse_trig_functions understanding-inverse-trig-functions	cosinus-og-sinus tangens

Topic	Khan Academy	Khan Academy exercises	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
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

Topic	Khan Academy	Khan Academy exercises	Webmatematik (Danish)
Defining derivatives	slope-of-a-line-secant-to-a-curve 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
Area calculations		area-between-a-curve-and-an-axis	bestemt-integral-og-areal areal-mellem-to-funktioner
Optimization	minimizing-sum-of-squares minimizing-the-cost-of-a-storage-container	optimization	optimering

Topic	Khan Academy	Khan Academy exercises	Webmatematik (Danish)
Introduction to differential equations	differential-equation-introduction finding-particular-linear-solution-to-differential-equation		hvad-er-differentialligninger gore-prove
Separable differential equations	separable-differential-equations-introduction particular-solution-to-differential-equation-example		losninger-til-differentialligninger eksponentiel-vakst inhomogene-lineare-forsteordens-differentialligninger separation-af-variable
Modeling	newtons-law-of-cooling

Topic	Khan Academy	Khan Academy exercises	Webmatematik (Danish)
Rules of integration and indefinite integrals	antiderivatives-and-indefinite-integrals indefinite-integrals-of-x-raised-to-a-power antiderivative-of-hairier-expression basic-trig-and-exponential-antiderivatives	antiderivatives indefinite-integrals	stamfunktion ubestemt-integral integrerede-funktioner regneregler-for-integraler
Definite integrals and area determination	area-under-rate-net-change simple-riemann-approximation-using-rectangles riemann-sums-and-integrals connecting-the-first-and-second-fundamental-theorems-of-calculus definite-integrals-and-negative-area integrating-function-sums	area-under-a-rate-function-equals-net-change area-between-a-curve-and-an-axis evaluating-definite-integrals properties-of-integrals	bestemt-integral-og-areal areal-mellem-to-funktioner

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.

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. LÞs 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.

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.

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.

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.

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.

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.