**Exercise 1: Working with MATLAB Variables**

**The Fibonacci Sequence and the Golden Ratio**

**a. **Execute the following commands to create a vector F
containing the first 10 values of the *Fibonacci sequence*.

>> F = zeros(1,10);

>> F(1) = 1;

>> F(2) = 1;

>> for I = 3:10;

F(I) = F(I-1) + F(I-2);

>>end

**b. **Use a single vector PHI, to compute the ratio PHI(J) =
F(J+1)/F(J) for J = 1:9. The limiting value of PHI as J increases without bound
is called the *golden ratio*, *Φ*.

**c. **Plot PHI. What is the approximate value of the golden
ratio?

**d. **In a new figure, plot 1/PHI (blue open circles) and
PHI–1 (red stars) on the same axes.

**e.
**In a new figure, plot
PHI.^3 + PHI.^4 and PHI.^5 on the same axes.

**Exercise 2: Basic Statistics and Data Analysis**

**Historical Snowfalls **

** **

**a.
**Load the data in
snowfall.txt into the

MATLAB workspace. (Hint: >> doc textread)

**b.
**Display the
distribution of the data in a histogram.

Experiment with different numbers of bins and

observe the effect on the shape of the distribution. (Hint: >> doc hist). Plot the histograms for 5, 10, 15, and 20 bins in the
same figure using subplots. (Hint: >> doc subplot)

**c.
**Compute the mean *μ* and standard deviation *σ* of the data. (Hint: >> doc std)

**d.
**Make a function (normhist) that give you the normalize histogram (histogram
divided by the area under the curve of the histogram).

**e.
**Plot the normal
density given by

using the *μ* and *σ* from c. on top
of the normalized histogram from d. Note that the density curve has an area of
1, so to ÒfitÓ the normalized histogram.

** **

**Exercise 3: Working with MATLAB Fitting Functions**

**Dictyostelium Chemotaxis**

** **

Dictyostelium cells, when placed in an external
gradient, localize a key signaling protein (CRAC) to their leading edge. By
fluorescently tagging CRAC with GFP, we can visualize the leading edge of a
cell. I have measured the intensity of (CRAC-GFP) vs. angle for a single
dictyostelium cell and saved the data in dicty.xls file. For this particular
cell, the maximum GFP intensity does not align with the direction of the
external gradient.

**a.
**Load the intensity
vs. angle data (dicty.xls) into the MATLAB Workspace. Plot the data
on a linear scale using solid blue circular markers.

The plot resembles a cosine function with the form *L*+*P**cos(q- f), where *L*
is the average localization of GFP in the membrane, *P* is the amplitude of the cosine function and f is the angular offset.

**b.
**Write an M-file
function that takes your initial guess for (*L*, *P* and f) as input, calculates the function *L*+*P**cos(q- f) and plots the function. How good was your initial
guess?

**c. **Fit the data to the function in **b,** by minimizing the least-squares. what are the fitting
parameters? (Hint: >>
doc fminsearch)

** **

** **

**Exercise 4: ODE Solver** (Hint:>>doc ode23)

**Simple harmonic oscillator (Ball on a spring)**

Consider a simple harmonic
oscillator (such as a block on a spring) with no driving force, and no friction
so the net force is just. Using NewtonÕs second law . The acceleration, *a* is
equal to the second derivative of *x*, which is given by and therefore
equations of motion simplify to: , where .

**a. **Create a Matlab function, (function dydt = bouncingball(t,y)), which return a two dimensional vector dydt=[dxdt,dvdt], where dxdt=v; and dvdt = -(k/m)*x;

**b. **Integrate the equation of motion to get the position
and velocity as a function of time, by calling the built in Matlab ODE solver
ode23 or ode45 (Hint:>>doc ode23)

**c. **What kind of motion do you expect from
such a system?

**d. **Now imagine that there is a damping term proportional to the velocity, how do the equations of motion
change?