Math 229 FINAL EXAM Review: Fall Final Exam Monday December 11 ALL Projects Due By Monday December 11

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1 Math 229 FINAL EXAM Review: Fall Final Exam Monday December 11 ALL Projects Due By Monday December Problem 1: (a) Write a MatLab function m-file to evaluate the following function: f(x) = x2 cos 2 x x 3 + 4x + 1 (b) Write the MatLab commands to generate 500 evenly spaced numbers on the interval π/2 x 3π/2 (c) Write the MatLab commands to generate the following list of numbers: 100, 98, 96,... 0 (d) Write the MatLab commands to generate the following list of numbers: when n = /n 2, 4/n 2, 9/n 2, 16/n 2,..., 1 (e) Write the MatLab commands to generate the following list of numbers: 0.1, 0.01, 0.001,..., (f) Write down the MatLab commands to find ALL the roots (zeros) of the following polynomial WITHOUT graphing the polynomial. List ALL the roots. f(x) = x 6 13x 4 + x x 2 + 1

2 Math 229 FINAL EXAM Review: Fall Problem 2: Use MatLab to find the TWO intersection points of the functions x 6 and e x/10. List the x values of the 2 intersection points to 2 decimal places Hint: Use of logarithms may help! 3. Problem 3: Use MatLab to numerically investigate the following limit: lim x 3 x 2 9 x (a) Write a MatLab function m-file to easily compute f(x) (b) Write MatLab commands to automatically generate 6 values of x which approach -3 from both the left and right. (c) Report the results of the limit in the form of a table. Write the MatLab commands used to generate this table. What is the limit? EXTRA CREDIT: Use Calculus to analytically determine this limit. 4. Problem 4: We know that some funtion f(x) has as its derivative df/dx = f (x) = (x + 1) e x. We want to know some things about the graph of the function, f(x). (a) Write a MatLab function m-file to easily compute the derivative function df(x)/dx (b) Write MatLab commands to graph f (x) on the interval 4 x 2. (c) Using the graph of f (x) and the first derivative test, describe all the local extrema of the function f(x). (d) Use the difference quotient (with h = 0.01) of f (x) to numerically approximate the second derivative f (x). Write MatLab commands used to find and plot this quantity. (e) Use your graph of f (x) to find all points of inflection of f(x). (f) If f(0) = 2, use the information you have to sketch a graph of f(x).

3 Math 229 FINAL EXAM Review: Fall Problem 5: Consider the function f(x) = sin2 x 16 x 2 on the interval ( π, π). (a) Write down the MatLab commands used to create a MatLab function to evaluate f(x) easily. (ie: Write a MatLab function m-file) (b) Find the difference quotient of the function for h = 0.01 and plot it on the same graph as the function. Write down the MatLab commands to do this. (c) Find the second difference quotient for the function using h = Plot this. Write down the MatLab commands to do this. (d) Using the graphs, find all the local minima and maxima of the function on the interval. Write down the x and y coordinates of these points. (e) Using the graphs, determine all the intervals where f(x) is concave up. 6. Problem 6: Consider the following integral: x dx (a) Use MatLab to find the value of the definite integral numerically using Riemann Sums. First, write down ALL the commands to compute the integral using 10 rectangles. What is the value of the integral. (b) Recompute the sum using 100 rectangles. (c) Recompute the sum using 1000 rectangles. (d) What value does sum converge to as n? EXTRA CREDIT: Compute the integral analytically.

4 Math 229 FINAL EXAM Review: Fall Problem 7: For some reason, an engineer needs to support an upside down parabola (y = 4 x 2 ) with two beams starting from the point (0, 2). We want to find the points on the parabola which are closest to the support point, (0, 2) in order to have the smallest beams. (a) Use MatLab to draw the parabola and set up the problem. (b) With d equal to the length of a line from (0,2) to any point (x, y) on the parabola, write down an equation for d in terms of one variable: ie, find the equation for d(x). (Remember the equation for the distance between two points??) (c) Determine the range of x in d(x) and write MatLab commands to plot this function. (d) Graphically, find the value of x which minimizes d. Find this minimum value of d. 8. Problem 8: Given that f (x) = 3x2 cos(πx/8). Use Matlab to find all relative maxima and minima of x 2 +1 the function f(x) on the interval 10 x 10. Write down the values of x where f(x) has either a relative maxima or a minima and label which. Use the information about the derivative to sketch a graph of the function f(x) such that f(0) = 0.

5 Math 229 FINAL EXAM Review: Fall Problem 9: Find the point P (x, y) on the curve y = e x/3 that is closest to the point Q(5, 3). (a) Plot the curve y (in black) and the point Q(5, 3) (as a red STAR) on the same graph. Write down MatLab commands used. From this picture, estimate the location of the point on the curve that is closest to (5, 3). (b) What function d(x, y) gives the SQUARED distance from P (x, y) on the curve to Q(5, 3)? (c) Write the m-file for the function d(x) (depending only on x) that gives the SQUARED distance from P (x, y) on the curve to Q(5, 3). (d) Using d(x) from part (c), what is difquo for d(x) with h = 0.001? (e) Plot difquo for 0 x 5. For which x is d(x) = 0 (to three decimal places)? (f) What is P (x, y) on the curve? Give coordinates to three decimal places. (g) Why is this xvalue a minimum for d(x)? Apply the first derivative test: How does the graph cross the xaxis? 10. Problem 10: A student in an Mth229 section is bored silly with with the professor asking her to do multiple Reimann sum calculations changing the number of rectangles, n, and also the limits of integration (a, b) in order to approximate the integral: b a Having had enough, she decides to write a Matlab function that takes, as INPUT, the values of n, a and b and spits out (as OUTPUT) the corresponding Riemann Sum. (a) In the space below, write this magic MatLab function. (b) Use this function to calculate (ie take the limit, as n, of the Riemann sum) the integral: 5 (c) Use the same approach to estimate: 5 and (d) If we define the number A as: use your magic function to estimate A 2 (e) What do you think the true value of A 2 is? a A = lim, a a