A function f(x) which computes:

Examples

In [ ]: f(0)
Out[ ]: 1.1176470588235294

In [ ]: f(2)
Out[ ]: 0.07293440360502447

While you do not need to do the following, it is possible to implement the function f so that it can compute the function for array input arguments. If you do this, the examples above should work as shown, and in addition you should be able to do:

In [ ]: import numpy

In [ ]: x = numpy.array([0, 2])

In [ ]: f(x)
Out[ ]: array([ 1.11764706,  0.0729344 ])

For the following tasks, it is not necessary to support numpy array arguments x as shown here but you may find this useful to plot f(x) more conveniently.

Write a function make_multiplier(factor) that returns a function which takes an argument x and which should return factor * x.

Examples

In [ ]: g = make_multiplier(10)

In [ ]: g(1)
Out[ ]: 10

In [ ]: g(2)
Out[ ]: 20

In [ ]: g = make_multiplier(0.5)

In [ ]: g(10)
Out[ ]: 5.0

In [ ]: g(100)
Out[ ]: 50.0

A function integrate_f_from0(b) which computes (an approximation of)

where the function f(x) is as defined in the training part of this laboratory. The function integrate_f_from0(b) should return a floating point number.

A function find_max_f() which returns (an approximation of) the x for which f(x) takes the maximum value. Your function find_max_f should return a floating point number.

A function find_f_equals_1() which returns a float which is (an approximation of) the x for which f(x)=1. We are only interested in the solution where x is negative.

A function lin_int(xs, ys) which accepts sequences xs and ys of data as input, and returns a callable object f(x) which returns y(x) using linear interpolation between the points provided by the data xs and ys.

You can write your own interpolation function or use a library function for this.

Example

In [ ]: xs = [1, 2, 3]

In [ ]: ys = [10, 20, 15]

In [ ]: f = lin_int(xs, ys)

In [ ]: f(1)
Out[ ]: array(10.0)

In [ ]: f(2)
Out[ ]: array(20.0)

In [ ]: f(1.5)
Out[ ]: array(15.0)

In [ ]: f(2.5)
Out[ ]: array(17.5)

Write a function make_oscillator(frequency) that returns a function which takes a floating point value t to represent time, and returns sin(t * frequency).

Examples

In [ ]: osc1 = make_oscillator(1)

In [ ]: osc1(0 * math.pi)
Out[ ]: 0.0

In [ ]: osc1(0.25 * math.pi)
Out[ ]: 0.7071067811865475

In [ ]: osc1(0.5 * math.pi)
Out[ ]: 1.0

In [ ]: osc2 = make_oscillator(2)

In [ ]: osc2(0 * math.pi)
Out[ ]: 0.0

In [ ]: osc2(0.25 * math.pi)
Out[ ]: 1.0