Exercises Mathematics

Basic Mathematics

fractions

A5

 

Euler, (ln and basic e)

 

A6)   [ln(a hoch b)]/b

A7)  ln (c hoch d)

A8)

a) Get solution of x = ?

b) Put this solution in the equation you solved, and look if it matches.

A9) lnx = -B + lnC²

x = ?

A10) ln(a) – ln(b) + 1

Quadratic equations and equation systems with 2 unknowns

A11) ax² + bx + c = O. Write down the discriminant and calculate x1 and x2 using discriminant.

A12) Create an Excelfomula for the calculation of the quadratic equation above.

A13) 3x² + 4x + 1 = 0 searched x1 and x2, solve with Excel formula created above.

Also solve the following equations with this Excel formula.

A14) x² + 2x + 2 = 0

A15) x² + x = 0

A16) x² + 2x + 1 = 0

A17) x² + 2 = 0

A18) x² – 2 = 0

A19) Equationsystem with 2 unknowns

a1x + b1y = c1

a2x + b2y = c2

Write down the determinants and ist solutions x= ?, y = ? using determinants

A20) Create excelformulas for D, D1 and D2 and for the calculation of x and y.

A21) Solve  following equationsystems with 2 unknowns below.

2x + 3y = 4

4x + 5y = 6

searched are x = ? and y = ?

A22)

6x + 3y = 10

8x + 4y = 7

searched are x = ? and y = ?

A23)

6x + 15y = 5

2,4x + 6y = 2

searched are x = ? and y =  ?

 

Exercises on differential and integral calculus

B24)

1) G(u) = e sqr(u)*sinu
G‘(u) = ?

B25)

2) J(v) = v³ + 5v sqr(5) + 1
J‘(v) = ? and how big is the tangent slope at v = 3?

B26)

f(r) = ∫r2/(E – r)2 = (r2 – 2E2)/(E – r)) + 2E*ln(E – r)

Check by deriving the integrated function.

 

B27)

Function given is  f(x) = x³ – 2x² + 5

1. Draw the graph of the function using millimeter paper or a mathematical computer program: f(x) = x³ – 3x² + 5.

2. How many turning points does the graph of this function have and where are they located? x = ?, y = ?.

   Derive f (x) and calculate f ‘ (x) = 0 resp. the both turning points (Slope = 0) which you ‘ve just 1) determined graphically. Compare the results.

3. How many zero places does this function have and where does it or do they lie, f‘ (x) = 0?  Find the solution by inserting x in f (x). (Use Excel-programm) 

4. Integrate f (x) from the one zero place (y = 0) to the 1st turning point, determined in 2.

   In the natural sciences, functions are frequently encountered that cannot be defined mathematically or if a mathematical function exists, that function
   cannot be integrated.

   First calculate this area by adding up the small squares = 1 mm2 per square. Now calculate the definite integral of f (x) from the zero point to the

   1st turning point. = ∫ (0 / zero) f (x) = (x³ – 2x² + 5) dx

C28)

Integrate f(x) from exercise B)27 from the zero point (y = 0) to the first turning point.

In science, one often encounters functions that cannot be defined mathematically, or if a mathematical function exists, it cannot be integrated.

First calculate this area by adding up the small houses = 1 mm2 per house. Now calculate the definite integral of f(x) from the zero to the 1st point of inflection. = ∫(0/zero)f(x) = (x³ – 2x² + 5)dx

Exercises on the differential equations

D29)

For the nonlinear first-order differential equation: y’ = b(M – y)(N – y), M ≠ N

the following solution was found:

y = M + (N – M)/[1 – L*e high b(N – M)x]

Check this result by inserting it into this differential equation: Notice: First calculate y’ or dy/dx

 

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