# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 19

*Exercises*

Lucy Sonnenschein is riding on her bike for five hours. In the first two hours, she makes km and in the following three hours, she also makes km. What is her average velocity?

Prove the mean value theorem for differential calculus for differentiable functions

and a compact interval , using the mean value theorem of integral calculus (you do not have to show that the average velocity is obtained in the interior of the interval).

Determine the second derivative of the function

An object is released at time and it falls freely without air resistance from a certain height down to the earth thanks to the (constant) gravity force. Determine the velocity and the distance as a function of time . After which time the object has traveled meters?

Let , be a continuous function, and let be a primitive function for . Show that is a primitive function for .

Let , be a continuous function and let be a primitive function for . Show that is a primitive function for .

Let , be a continuous function and let be a primitive function for . Show that is a primitive function for .

Determine a primitive function for

whose value at equals .

Compute the definite integral

Compute the definite integral

Compute the area of the surface, which is enclosed by the graphs of and of .

Let be the minimal positive number fulfilling . Compute the area of the surface, which is enclosed by the graph of the cosine function and the graph of the sine function above .

Compute the definite integral of the function

on .

Determine the average value of the square root for . Compare this value with the square root of the arithmetic mean of and and with the arithmetic mean of the square root of and of the square root of .

Show that for every the estimate

holds. Hint: Consider the function on the interval .

Determine for which the function

has a maximum or a minimum.

A person wants to sun bath for an hour. The intensity of the sun in the time interval (in hours) is given by the function

Determine the starting point for the sun bath in order to get the maximal amount of sun.

According to recent studies, the student's attention skills during the day are described by the following function

Here, is the time in hours and is the attention,n measured in micro-credit points per second. When should one start a one and a half hour lecture, such that the total attention skills are optimal? How many micro-credit points will be added during this lecture?

Let be a differentiable function and let be a continuous function. Prove that the function

is differentiable and determine its derivative.

Let be a continuous function. Consider the following sequence

Determine whether this sequence converges and, in case, determine its limit.

Let be a convergent series with for all and let

be a Riemann-integrable function. Prove that the series

Let be a Riemann-integrable function on with

for all . Show that if is continuous at a point with , then

Prove that the equation

has exactly one solution .

Let

be two continuous functions such that

Prove that there exists such that .

Let

be a continuous function with

for every continuous function . Show .

*Hand-in-exercises*

### Exercise (3 marks)

Compute the definite integral , where the function is

### Exercise (3 marks)

Compute the definite integral

### Exercise (2 marks)

Determine the area below the graph^{[1]}
of the sine function between
and .

### Exercise (3 marks)

Determine an antiderivative for the function

### Exercise (4 marks)

Compute the area of the surface, which is enclosed by the graphs of the two functions and such that

### Exercise (3 marks)

Let

be two continuous functions and let for all . Prove that there exists such that

*Footnotes*

- ↑ We mean the area between the graph and the -axis.

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