MAE 284 Quiz 1-6, Exam 1-2 Conceptual Review
True
Q6 MATLAB's gradient function takes a vector of length n and returns a vector of length n. The first element is a forward difference while the last element is a backwards difference. The remaining elements are central differences. (t/f)
True
Q6 Numerical differentiation is the process of approximating a continuous derivative by an algebraic equation. (t/f)
True
Q6 Partial differential equations have more than one independent variable. (t/f)
False
Q6 The finite difference method is based on guessing a value for one of the initial conditions and using an initial value algorithm.
False
Q6 The finite difference method is used to solve initial value problems.
False
Q6 The numerical approximations of the derivatives discussed in class were derived from Newton-Raphson theory.
False
Q6 The only way to increase the accuracy of a numerical approximation of a derivative is to decrease the step size h.
Gauss-Seidel
Q6 What technique did we use to solve nonlinear finite-difference equations?
Gauss-Seidel
Q6 What technique did we use to solve nonlinear finite-difference equations? a. Gauss-Seidel b. Linear Interpolation c. Something Else
Robin, Neumann, Dirichlet
Q6 Which of the following are types of boundary conditions we discussed in class?
True
Q6 Initial value problems have all necessary conditions given at a single value of the independent variable. (t/f)
opposite
Q2 Bracketing methods are based on the idea that if two function values, and , have ________ signs, there must be at least one root between them.
True
Q2 For the fourth iteration in the Newton-Raphson method, you will use the estimated root location from the third iteration as the new initial point. (T/F)
[p,r] = mae284(a,b,[ ],d)
Q2 Suppose you are given a function with the following inputs and outputs. [p,r] = mae284(a,b,c,d) What should the line of code read if you wish to use the default input for c? (You want to specify a,b, and d only) Default inputs are defined in the function "mae284" in case the user does not specify them.
False
Q2 The Newton-Raphson method is based on determining the intersection of the tangent of the function and the x-axis. (T/F)
True
Q2 The bisection method is a bracketing method. (T/F)
True
Q2 The golden section method helps to minimize computational costs in finding a maximum/minimum.
True
Q2 The main advantage of the bisection algorithm is that it always converges. (T/F)
True
Q2 The roots of a function f(x) are located at the points where the function crosses the x-axis. (T/F)
True
Q2 The secant method is similar to the Newton-Raphson method, but uses an approximation of the derivative. (T/F)
4
Q3 A single application of Simpson's 3/8 Rule requires computing the value of the function at how many points?
False
Q3 Fubini's Theorem states that the order of integration is important. (t/f)
True
Q3 In general, the Composite Trapezoid Rule is more accurate than a single application of the Trapezoid Rule. (t/f)
False
Q3 Romberg's method can be used to combine two O(h4) estimates to obtain a single O(h8) estimate.
straight line
Q3 The Trapezoidal Rule approximates the function by a ______________.
divisible by 3
Q3 The composite Simpson's 3/8 Rule requires the number of segments to be ________.
True
Q3 The trapezoid rule can be derived from the method of undetermined coefficients.
True
Q4 | [a b; 2a 2b] |=0
False
Q4 | a b | | 0 0 | =a (t/f)
True
Q4 At the start of Gauss Elimination, the first equation is known as the pivot equation and the a11 element is known as the pivot element. (t/f)
True
Q4 Gauss Elimination is the systematic process of removing unknowns using forward elimination followed by back substitution to obtain the unknown values.
False
Backward finite-difference approximations require the function value, f(xi), at the point at which the derivative is to be calculated, ( xi, f(xi) ) and function values to the right (e.g. f(xi+1), f(xi+2), ...).
False
Q4 If the determinant of a matrix is zero, the inverse of the matrix is a scalar. (t/f)
True
Q4 If two rows of a matrix are identical, the determinant is zero.
True
Q4 Naïve Gauss Elimination fails if the pivot element is zero.
[grade] =
E1 A TA is creating the following function header, but can't remember how to specify an output. Click the dropdown menu and select the correct component. function [grade] = autoGrader(aNumber, answersArray)
True
E1 Bracketing methods always require two initial estimates. (T/F)
False
E1 In the golden section method, if f(x1) < f(x2), then the new search interval is from xlow to x1. (t/f)
False
E1 Inflection points and local maximums and minimums greatly increase the speed at which the Newton-Raphson method converges. (T/F)
True
E1 MATLAB's built-in fzero function is uses a combination of bisection, secant, and inverse quadratic interpolation methods and a search method if only one input is provided . (T/F)
True
E1 MATLAB's mesh function can be used to visualize a function of two variables. (t/f)
Bisection and False Position
E1 Select the numeric root finding methods that use bracketing: Bisection Newton-Raphson Secant Modified Secant False Position Incremental Search
A^(-1) inv(A) A^-1
E1 Suppose you are given a matrix A, which command/s will calculate the inverse of A? Select all those that apply. A^(-1) inv(A) A^-1 inverse(A)
if nargin < 4, s = 100; end
E1 Suppose you are given the following function: [a,b] = mae284(p,q,r,s) If the user does not specify the input value for 's', which line of code can be used to define a default value of s = 100 within the function?
True
E1 Tangent points in a function can cause an opposite-valued bracket to contain an even number of roots. (T/F)
False
E1 The bisection method is most effective when there are multiple roots inside a bracket. (T/F)
zero
E1 The first derivative of a function must be zero at a maximum.
zero
E1 The first derivative of a function must be zero at a minimum.
Evenly
E1 The incremental search method uses input values spaced at evenly -spaced intervals.
False
E1 The main advantage of the Newtone-Raphson Method is that it always converges. (t/f)
True
E1 The main disadvantage of the Newton-Raphson method is that it may not converge. (T/F)
Graph the function
E1 What is the first step if you are asked to find all real roots of a function?
importdata
E1 Which of the following functions will import a data file (for example, from a .txt file)? a. import-data b. importdata c. import_data d. import data
x = 0:0.1:10
E1 Which one of the following commands generates an array of equally spaced elements?
4
E2 A single application of Simpson's 3/8 Rule requires computing the value of the function at how many points?
True
E2 A system of equations can be written in the form: Ax=b, where A is a square matrix, x is a vector of unknowns, and b is a vector of constants. (t/f)
True
E2 Adaptive Quadrature uses more segments in regions where the function changes rapidly. (t/f)
True
E2 An upper triangular matrix is an echelon matrix. (t/f)
16
E2 How many points would we need to determine a double integral using Simpson's 3/8 rule?
True
E2 Ill-conditioned systems result from equations that correspond to parallel lines, coincident lines, or lines with nearly the same slope (nearly parallel). (t/f)
True
E2 Romberg's method can be used to combine two O(h2) estimates of the integral to obtain a single O(h4) estimate.
False
E2 Simpson's 1/3 Rule can be used to calculate the integral of a function or data with unequal-width segments. (t/f)
second-order polynomial third-order polynomial first-order polynomial
E2 Simpson's 1/3 Rule will result in zero error for a ______.
third
E2 Simpson's 3/8 Rule approximates the function by a ________ -order polynomial
False
E2 The Composite Simpson's 1/3 Rule can be used to obtain the integral over an interval divided into n=5 segments.
False
E2 The Composite Simpson's 3/8 Rule can be used to obtain the integral over an interval divided into n=5 segments. (t/f)
True
E2 The determinant of ill-conditions systems is zero or nearly zero. (t/f)
False
Q4 Naïve Gauss elimination ensures that division by zero is not possible. (t/f)
True
Q5 A single third order ODE can be converted into a system of 3 first order ODEs. (t/f)
True
Q5 In the equation below, h is the step size. y(i+1)=y(i)+PHI*h
False
Q5 In the following ODE, y is the independent variable. dydx=5y+e−x
False
Q5 Using an explicit method on a stiff system will, in general, take less time to run than an implicit method. (t/f)
Truncation Errors
Q5 Which of the following are caused by using numerical methods to approximate an exact solution? a. Round-off errors b Truncation errors
Euler
Q5 Which of the following is based on using only the slope of the solution of the ODE at the current point to calculate the value of the solution at the next point? a. Heun b. Midpoint c Euler
Euler
Q5 Which of the following is the simplest Runge-Kutta method?
False
Q6 Eigenvalues are related to the mode shape.
True
Q6 Eigenvalues are the solutions of the equation: | A- λ*I |=0 (t/f)
True
Q6 For non-linear ODEs, we can use the shooting method along with a root finding method to drive the error in the given boundary condition to zero. (t/f)
Centered
Q6 If we only have two function values, which type of method will result in the best approximation of the derivative? a. backward b. forward c. centered
True
Q6 If λ1 is an eigenvalue associated with eigenvector v1, then A*v1 = λ1*V1. (t/f)
True
Q6 Increasing the damping decreases the magnitude of the response at the resonance frequency.
True
Q4 An upper triangular matrix is an echelon matrix. (t/f)
True
Q6 Initial value problems have all necessary conditions given at a single value of the independent variable.
function
Q1 A computer scientist is attempting to automate the process of cooking hamburgers. They are using MATLAB to analyze some data, and need a function file that does not have any output variables. Click on the dropdown menu to complete the header. function flippingBurgers(burgerTemperature, juiceIndex, condimentArray)
function [d, e] = quiz1_question4(a, b, c)
Q1 A student is attempting to create a function file named quiz1_question4.m After three tries, they type a function header that works correctly. All four attempts are listed below. Which one of the listed function headers is error-free? function [d, e] = quiz1_question4(a, b, c) function [d, e] = boss_function(a, b, c) quiz1_question4(a, b, c) = function [d, e] quiz1_question4(a, b, c) = [d, e]
GPA_no_skipping
Q1 A student wants to skip the next three MAE 284 labs, and is writing some MATLAB code to see what that will do to their GPA. They need to name a variable, but for some reason can't remember which names are valid. Of the four potential variable names below, which one is valid? GPA-3-skipped-labs GPA_no_skipping GPA no skipping 3_skipped_labs_GPA
False
Q1 All mathematical problems have a closed-form solution (i.e. they can be solved for a variable without having to make guesses). (T/F)
False
Q1 An analytical solution is an approximation.
3*(4^2)
Q1 Choose the statement that is equivalent to the expression below: 3*4^2 a. (3*4)^2 b. 3*(4^2) c. 4*3^2
cross(a,b)
Q1 Given two vectors a and b, which command would you use to calculate the cross product of a and b?
False
Q1 Having problems with retrieving or submitting homework are valid excuses for late submission. (t/f)
False
Q1 In your homework, you can get partial credit for your work after an error. (t/f)
True
Q1 The equal sign, "=", is used to assign data to a variable. (T/F)
1)no 2)no 3)yes 4)yes
Q1 The equation below is being evaluated in MATLAB. For each labeled operation, indicate whether it requires a dot operator. R and T are scalars, and P, A and V are vectors of the same length. mdot=p(1)/(r(2)*t)(3)*a(4)*v
True
Q1 Vectors are transposed using the single quote operator. (T/F)
factorial(m)
Q1 Which command can be used to calculate the factorial of m?
surfc
Q1 Which command will generate a surface plot with contours?
negative
Q1 Will the following statement be positive or negative? -4^2
False
Q1 You are not allowed to discuss and work as a group in solving assignments.
False
Q1 You can have your phone on your desk during class, lab, quizzes, or exams.
True
Q3 If the error of an integration technique is proportional to h2, we say the error is of order h2, which we write as O(h2). (t/f)
True
Q3 Integration is equivalent to finding the area under a function for positive functions.
True
Q3 Numerical integration can be used to calculate the integral of discrete data points.
True
Q3 Two applications where numerical integration is used are integrating discrete data points and when it is difficult or impossible to obtain an analytic solution. (t/f)
incsearch
Which function can be used to find the number of roots of a function between a specified lower and upper bound?