8/13/2019 schriftliche Aufgabe

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Task 1

a) Give the maximum each domain and the corresponding set of values of the function and sketch

the graph.

b) Where is the line

i ) Add to the equation of the straight line h , that is parallel to and passes through g P ( 2/2) !

ii ) Add the equation of the line s , which is perpendicular to and passes through g P ( 2/2) !

iii ) Explain in your own words , what is the connection between parallel and perpendicular linesthere .

iv ) Draw the lines g , h and n in a coordinate system !

c ) The point Q ( 2 / f ( x)) to the function f (x) = 1/x a straight s down to the function f

perpendicular . Determine the equation of the perpendicular line n and compute its zero ! ( Drawing

is not necessary.)

d)

i) a straight line g : y = mx + c is reflected in the y -axis.

What is the relationship between the slope of the original line and the slope of the mirror line ? Give

reasons for your answer!

ii) a straight line g : y = mx + c is in the first Median ( line y = x ) mirrored .

What is the relationship between the slope of the original line and the slope of the mirror line ? Give

reasons for your answer!

Task 2

Given the function

a) Examine the function f !

i) symmetry

ii ) Asymptotic behavior and asymptotic function

iii ) zeros

iv ) extreme values ( necessary and sufficient condition )

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v ) Turning Points ( necessary and sufficient condition ) ( no turning tangent !)

b ) Draw the function f with all the characteristic points in the interval [ -1.5,5 ]; calculate 5LD edge

spots!

c) The cross section of a mountain range in math land is by the function f in the interval [-1,4]; given .

One unit on the x- and y- axis corresponds to each 1000 m.

( Also note the new boundary points . )

i) In the mountains ( graph of the function f) at the points x = 1 and x = 2 ever built a mountain cabin.

What is the difference in height between the two mountain huts ?

What is the distance between the two mountain huts in a straight line ?

ii) Give the highest and lowest point of the mountain ! ( In the interval [-1 , 4 ] !)

iii ) At what point x the mountain is steepest ?

iv) Math country ski runs are only up to a slope m of the slope between - 0.5 and + 0.5 allowed .

(1 ) Which of pitch angle corresponds to m = 0.5 ?

(2 ) In what subintervals of [-1 , 4 ] ski slopes are allowed in these mountains ?

Task 3

a) Some telescopes will be set a correcting lens in front of the concave mirror .

The upper part of the cross section of such a lens is approximated by the function f(x) = 0,5 x4 - 0,5 x

2 

described .

The lower part of the cross section corresponds to a constant function.

At which two points ( only x - coordinate) is the cross section of the thinnest lens ? Document all

steps necessary computing !

b ) A ball starts at A (- 2 / 0), and rolls along a parameter- shaped path . At point B, the path ends at

this point and the ball rolls on a tangential line and reaches the point N (5/ 0) , the x -axis.

The parameter -shaped path described by a portion of the graph of the function f (x) = 1/2 x2 - 2.

Determine with suitable mathematical methods, the coordinates of the point B at which the ballleaves the web.