1
Numerical differentiation formulas for a function f are typically found by doing which of the following?
 A. Finding a zero of f' B. Approximating f by a polynomial P and differentiating P C. Differentiating a Fourier transform of f and then applying an inverse transform D. Finding the slope of f at two randomly generated points E. None of the above
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Question 2
Let . Approximate using the 2-point difference formula with .
 A. 2.0 B. 2.1 C. 1.9 D. 0.21 E. 0.19
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Question 3
For fixed , , and finite difference formula, there is an that minimizes the error in the computed approximation to . Why?
 A. Every difference formula has a truncation error formula that can be minimized. B. High degree interpolating polynomials oscillate too much. C. Underflows increase as . D. Overflows increase as . E. Rounding errors increase as .
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Question 4
Use a 3-point centered difference formula with to approximate , where
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 A. 2.000 B. 1.999 C. 1.989 D. 1.899
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Question 5
If f(x) is only known at the uniformly spaced grid points and a approximation to is desired at each , then what combination of differentiation rules could be used?
 A. Centered difference formula at each B. 3-point difference formulas, forward difference at left endpoint, backward difference at right endpoint, and centered difference elsewhere C. Backward difference on left half of gridpoints, and forward difference on right half D. Alternating forward and backward differences, beginning with backward difference at
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Question 6
What is the truncation error term for the 3-point centered difference formula for approximating ?
 A. B. C. D.
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Question 7
Let . Approximate using the 2-point difference formula with .
 A. 2.0 B. 2.1 C. 1.9 D. 0.21 E. 0.19
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Question 8
Which of the following statements about composite Newton-Cotes quadrature rules is true?
 A. They use a Taylor polynomial for f'. B. They integrate a piecewise polynomial approximation to f. C. They are a type of Monte Carlo method. D. They require orthogonal polynomial integration. E. None of the above
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Question 9
Use Simpson's method () to approximate . Choose the best 4 significant digit answer below.
 A. 0.3750 B. 0.5000 C. 0.6667 D. 0.3333 E. 0.2500
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Question 10
Use composite trapezoidal rule with to approximate . Choose the best 4 significant digit answer below.
 A. 0.3750 B. 0.5000 C. 0.3333 D. 1.500 E. 0.3125
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Question 11
 A. Subdivides if the integral is too large. B. It subdivides if the derivative is too large. C. It subdivides if rounding error exceedes truncation error. D. It subdivides if error estimate is too large.
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Question 12
Newton-Cotes quadrature rules for a function f are typically found by doing which of the following?
 A. Using a left sided Riemann sum B. Approximating f by a polynomial P and integrating P C. Integrating a Fourier transform of f and then applying an inverse transform D. Finding the average height of f at n randomly generated points E. None of the above
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Question 13
In a floating point system with 3 decimal digits in the fractional part, what is the result of 7.234 + 5.784?
 A. 13.02 B. 13.01 C. 13.0 D. 13.1 E. None of the above
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Question 14
What is the relative error in the 3 decimal digit floating point arithmetic computation (123 + 0.449)-122?
 A. 31% B. 0.449 C. 44.9% D. 0 E. 1.449
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Question 15
What value results in the 3 decimal digit floating point arithmetic computation (123.3 + 0.4)-122.7?
 A. 1 B. 0.4 C. 0 D. 0 E. 1.3
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Question 16
In a floating point system with t digits in the fractional part, if x and y share the same normalized exponent and the same g fractional digits, then how many digits of fl(x-y) are known to be correct?
 A. t-g B. g C. All t D. 0 E. It depends on the compiler.
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Question 17
In general, which of the following holds for the floating point addition of two or three floating point numbers whose sum does not overflow or underflow?
 A. Both commutative and associative B. Commutative but not associative C. Neither commutative nor associative D. Associative but not commutative
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Question 18
Swamping and cancellation both describe a loss of information. Select the statement below that best describes their effects.
 A. Swamping does not violate the fundamental axiom of floating point arithmetic, but cancellation does. B. Cancellation loses precision, while swamping does not. C. Swamping loses precision, while cancellation does not. D. Swamping can only happen with multiplication and cancellation only with addition.
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Question 19
The machine epsilon is nearest to which of the following?
 A. Underflow B. Overflow C. The distance between 1 and the nearest float to 1 D. 1/Overflow
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Question 20
Suppose x and y are floats and fl(x+y)=fl(x). What can be said of y?
 A. y underflowed. B. y=0. C. |y| is less than |x|*(machine epsilon). D. fl(0+y)=0.
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Question 21
With a fixed length floating point word, adding a bit to the fractional part requires removing a bit from the exponent. Which of the following statements describes how this affects the set of floats?
 A. The floats are farther apart but have a larger range. B. The floats are nearer each other but have a smaller range. C. There are more floats. D. Underflow is smaller. E. The machine precision grows.
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Question 22
Suppose a floating point system can represent K floats. If one bit is added to that floating point word, about how many floats can then be represented?
 A. B. C. D. E.
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Question 23
What can be said of any solution, x, to fl(1+x)=1?
 A. . B. . C. . D. .
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Question 24
Fill in the blank. If it can be shown that the result of a computation is the exact answer to a nearby problem, then we say the computation is ____________.
 A. Highly accurate B. Rounding correct C. Backward stable D. Well conditioned E. Robust
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Question 25
If x + y is smaller than both x and y, information in the sum x + y is lost. What is this called?
 A. Swamping B. Machine epsilon C. Truncation error D. Cancellation E. None of the above
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Question 26
Fill in the blank. If a small change to a problem leads to a large change in its solution, then we say that the problem is ______________.
 A. Backward stable B. Well conditioned C. Robust D. Ill conditioned E. Highly accurate
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Question 27
Problem has input and output . A small perturbation gives a new input and a new output . A relative condition number for problem should satisfy which of the following?
 A. B. C. D.
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Question 28
Problem has input and output . A small perturbation gives a new input and a new output . An absolute condition number for problem should satisfy which of the following?
 A. B. C. D.
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Question 29
Under what conditions on a problem and method are we guaranteed a good solution?
 A. Problem is stable and method is backward stable. B. Problem is stable and method is well conditioned. C. Problem is well conditioned and method is well conditioned. D. Problem is well conditioned and method is backward stable.
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Question 30
What is an upper bound on the relative error between two neighboring floats called?
 A. Cancellation limit B. Machine epsilon C. Underflow D. Backward error E. None of the above
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Question 31
In a floating point system with 3 decimal digits in the fractional part, what floating point number would be used to represent 0.0032419?
 A. 0.003 B. 0.00324 C. 0.003242 D. 0.00 E. None of the above
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Question 32
In a floating point system with 3 decimal digits in the fractional part, what floating point number would be used to represent 142.32419?
 A. 142 B. 142.324 C. 142.3 D. 142.32 E. None of the above
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Question 33
When a large magnitude and small magnitude float are added, information from the smaller number is lost. What is this called?
 A. Chopping B. Cancellation C. Truncation Error D. Swamping E. Tail Error
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Question 34
The term Initial Value Problem is best defined in this course as which of the following?
 A. , where with initial value B. C. Given , find . D. Find such that . E.
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Question 35
Which of the following is an initial value problem?
 A. . B. . C. . D. .
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Question 36
A shooting method for a boundary value problem requires what 2 numerical methods?
 A. An interpolator and a numerical differentiation rule B. A quadrature rule and an IVP solver C. A root finder and a quadrature rule D. An IVP solver and a root finder
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Question 37
What is the Euler slope?
 A. B. C. D. E. None of the above
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Question 38
In theory, the error in Euler's method can be made arbitrarily small by varying h, why is this not the case in practice?
 A. Because of rounding errors B. Because of truncation errors C. As , the truncation errors explode. D. As , the truncation errors explode.
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Question 39
When converting a scalar differential equation of order n into a system of first order differential equations, how many dimensions are needed to represent the solution vector?
 A. B. C. D. E. None of the above
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Question 40
Why does an explicit 3-step method for IVP require the use of a single-step method?
 A. Because it averages 3 values in B. Because Euler's method is unstable C. Because the corrector needs a prediction D. Because a 3-step method needs 2 previous approximations to y E. Because a 3-step method needs 3 previous approximations to y
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Question 41
An explicit 3 step method needs to use a single step method, but Euler's method is a bad choice. Why shouldn't one use Euler's method in this setting?
 A. Euler's method is unstable. B. Eulers method has local truncation error , which is lower than a 3 step method. C. Euler's method has large rounding errors. D. Euler's method is too slow for a 3 step method. E. Taylor methods are more general.
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Question 42
In a multistep predictor-corrector method, what type of method is the corrector?
 A. The corrector is typically a higher order Runge Kutta method. B. The corrector is typically a low order Runge Kutta method. C. The corrector is typically an implicit method. D. The corrector is typically a higher order explicit multistep method.
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Question 43
The Adams-Bashforth 2-step (AB2) method has the iteration . Consider the IVP . Select the best 4 significant digit approximation below to the AB2 value for , taking i, and .
 A. 1.297 B. 1.015 C. 0.7650 D. 1.547
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Question 44
How many function evaluations are required per timestep for an Adams-Bashforth method of order ?
 A. 1 B. 2 C. 4 D. 5 E. 6
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Question 45
Which of the following is an implicit method?
 A. B. C. D.
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Question 46
Which of the following is a general purpose single step method?
 A. Taylor's method B. Adams-Bashforth's method C. Adams-Moulton's method D. Runge-Kutta's method
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Question 47
Consider the IVP . Approximate using Euler's method with . Select the best 3 significant digit answer below.
 A. 3.00 B. 3.75 C. 2.00 D. 4.55 E. 6.30
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Question 48
Variable step-size methods all require which of the following?
 A. A good starting guess B. An error estimate C. Multistep method D. A list of allowable step sizes
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Question 49
Choose the iteration below which represents the Runge-Kutta method called the midpoint method.
 A. B. C. D.
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Question 50
Choose the iteration below which represents the Runge-Kutta method called the modified Euler or Heun's method.
 A. B. C. D.
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Question 51
How many function evaluations are required per timestep for a Runge-Kutta method of order ?
 A. 1 B. 2 C. 4 D. 5 E. 6
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Question 52
How many function evaluations are required per timestep for a Runge-Kutta method of order ?
 A. 1 B. 2 C. 4 D. 5 E. 6
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Question 53
Consider the IVP . Which of the following iterations correspond to the Taylor method of order 2 for this IVP?
 A. B. C. D.
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Question 54
If an IVP is stiff, what restriction is imposed on the solver?
 A. Stiff IVP's require a small timestep. B. Stiff IVP's require predictor-corrector methods. C. Stiff IVP's require Taylor methods. D. Stiff IVP's require high order methods.
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Question 55
What qualitative property should a method have when applied to the test problem ?
 A. . B. bounded for all . C. bounded for all . D. .
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Question 56
Why are Taylor methods not as general purpose as Runge-Kutta methods?
 A. Taylor methods require the evaluation of for \$k>1\$. B. Taylor polynomials oscilate too much. C. Taylor methods replace derivatives with function evaluations. D. Taylor methods are not parallelizable because of nested function evaluations.
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Question 57
Fill in the blank. Ignoring rounding errors, a successful IVP method with local truncation error of order k should have global error of order ________.
 A. k+1 B. k/2 C. 2k D. k-1
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Question 58
All single step IVP methods have a common iteration . If , what is the value of that would guarantee was on the solution curve?
 A. . B. . C. . D. .
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Question 59
What conditions on an initial value problem guarantee a unique solution?
 A. is smooth enough for all . B. is continuous in . C. is nonsingular. D. The initial is close enough to the solution. E. has a fixed point such that .
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Question 60
Fill in the blank. A second order ordinary differential equation, requires two extra conditions for uniqueness of solution. If these conditions are and , then the differential equation is called a(n) ___________.
 A. Partial differential equation B. Ill posed differential equation C. Side-condition differential equation D. Boundary value problem
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Question 61
Which of the following gives sufficient conditions for to have a unique solution?
 A. . B. continuous in , and exists in . C. continuous in , and continuous at . D. well-defined at the critical point .
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Question 62
The Weierstrauss approximation theorem says that under certain conditions on and for any , there is a polynomial , such that for all . Choose from below the weakest such conditions on ?
 A. has a power series representation on . B. is continuous on . C. has one continuous derivative on . D. has no poles on .
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Question 63
Which of the following is true for ?
 A. B. C. D.
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Question 64
Which of the following is true for ?
 A. for near . B. for near . C. for near . D. for near .
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Question 65
What is the minimum number of multiplications required to evaluate for arbitrary real ?
 A. B. C. D.
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Question 66
When dividing a polynomial of degree by the polynomial , the remainder will be a scalar. What choice below best describes this scalar?
 A. It has degree or degree . B. It is . C. It is . D. It is a root of . E. It is deflated.
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Question 67
Let be a zero of a polynomial of degree n > 10, and suppose we have an initial approximation such that both Newton's method and the secant method both converge, and an interval of length about containing and for which bisection will converge to . For this specialized case, list from fastest to slowest, the three methods.
 A. Newton's, secant, bisection B. Newton's, bisection, secant C. Secant, Newton's, bisection D. Secant, bisection, Newton's E. Bisection, Newton's, secant
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Question 68
Fill in the blank. If is a root of a polynomial of degree , then is a polynomial of degree . Finding the remaining roots of by finding the roots of is called ______________.
 A. The Chinese remainder method B. Degree slashing C. Synthetic division D. Deflation
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Question 69
Why don't we use the quintic equation to find the zeros of a polynomial of degree 5 or less?
 A. Newton's method is more accurate. B. Extracting roots is very slow on a computer. C. It is pretend. D. Rounding errors and truncation errors work against each other.
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Question 70
A Hermite interpolator is constructed for values and slopes at 4 knots. What can be said about the degree of this interpolator?
 A. Its degree is 4. B. Its degree is no more than 4. C. Its degree is no more than 7. D. Its degree must be more than 7. E. None of the above
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Question 71
What type of polynomial interpolator would be appropriate for the data ?
 A. Hermite B. Taylor C. Lagrange D. Piecewise linear E. Quadratic spline
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Question 72
If is the Lagrange interpolator for the knots and , what is ?
 A. 2.6 B. 2.8 C. 3.9 D. 1.5 E. 2.9
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Question 73
Suppose we approximate over using the Lagrange interpolator for the nodes and . Use the Lagrange error term (remainder) to get an upper bound on the error.
 A. B. C. D. E. There is no error; it is well posed.
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Question 74
How many polynomials exist which interpolate the data where for ?
 A. Only one B. Depends upon the knot positions C. n+1 Lagrange basis functions D. Infinitely many
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Question 75
Suppose is the Lagrange interpolator for the knots with for . If is a polynomial of degree satisfying , for . What can be said of ?
 A. B. C. D. is a Hermite interpolator. E. None of the above
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Question 76
If is the Lagrange interpolator for the knots and , what is ?
 A. 4.1 B. 3.8 C. 3.9 D. 3.5 E. 2.9
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Question 77
A Taylor polynomial is a special case of what type of interpolator?
 A. Osculating polynomial B. Lagrange interpolator C. Hermite interpolator D. Vandermonde interpolator E. None of the above
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Question 78
Consider the Lagrange basis functions for the knots , and . How many basis functions are there for this set of knots, and what are their degrees?
 A. Three basis functions, each of degree 2 B. Two basis functions, each of degree two C. Two basis functions, each of degree three D. Three basis functions, of degree 1, degree 2 and degree 3
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Question 79
Let be the value of the derivative of a curve at the point . Suppose we want to interpolate the data and . If a Vandermonde matrix is constructed for the interpolating polynomial, what is its size?
 A. B. C. D.
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Question 80
Let be the value of the derivative of a curve at the point . Suppose we want to interpolate the data and . In general, what is the degree of such an interpolating polynomial?
 A. 4 B. 5 C. 6 D. 7
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Question 81
If is smooth enough in a neighborhood of , then the Taylor polynomial of degree n for at approximates with error_________.
 A. B. C. D. E. None of the above
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Question 82
What is the Taylor polynomial of degree 1 for at ?
 A. B. C. D. E.
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Question 83
What is the Taylor polynomial of degree 1 for at ?
 A. B. C. D. E.
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Question 84
The linear Taylor polynomial at for is . Use this Taylor polynomial to approximate . Select the best 5 significant digit answer to the Taylor approximation.
 A. 10.149 B. 10.150 C. 5.5074 D. 9.8500 E. 10.001
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Question 85
The set of polynomials of degree n over a field is a vector space of which dimension?
 A. n B. 2n C. n+2 D. n/2 E. None of the above
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Question 86
The method of bisection can find a zero of a continuous function, if which of the following is true of the initial data?
 A. It is close to the correct answer. B. It gives the bisector of the zero. C. It is not complex. D. It gives function height and slope. E. None of the above
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Question 87
Bisection is used on an initial interval [a,b], where b-a=2. How many iterations are required for the current interval to have length 1/16?
 A. 4 B. 5 C. 6 D. 8 E. 10
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Question 88
If the method of bisection is used to find a zero of a continuous function f(x), and an initial interval [a,b] is given such that f(a)f(b) < 0, counting multiplicity, what can be said about the zeros of f in [a,b]?
 A. f has exactly one zero in [a,b]. B. f has 0, 1, or infinitely many zeros in [a,b]. C. f has an even number of zeros in [a,b]. D. f has an odd number of zeros in [a,b].
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Question 89
Which method cannot be used to find a root of ?
 A. Newton's method B. Bisection method C. Secant method D. Meuller's method E. False position
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Question 90
Suppose it is known that the continuous function has exactly 1 root strictly between 0 and 1, and it is not a double root. Which method can guarantee an approximate root with error less than 0.0001 in fewer than 14 function evaluations?
 A. Newton's method B. Secant method C. Bisection method D. None of the above
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Question 91
Suppose it is known that the continuous function has exactly 1 root strictly between 0 and 1, and it is a double root. An encrypted subroutine will return given any . Which of the following methods cannot be applied to this problem?
 A. Neither Newton's nor bisection can be applied here. B. Neither Newton's nor secant can be applied here. C. Bisection cannot be applied here. D. Newton's method cannot be applied here.
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Question 92
Which of the following gives a rough estimate of convergence for bisection and Newton's method (in the case each converge at their respective order of convergence)?
 A. For each iteration, Newton's method adds 1 correct bit and bisection add about 0.5 correct bits. B. For each of the two iterations, Newton's method doubles number of correct bits and bisection adds 2 correct bits. C. For each iteration, Newton's method doubles the number of correct bits and bisection adds 1 correct bit. D. For each iteration, Newton's method adds two correct bits and bisection adds 1 correct bit.
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Question 93
Suppose Newton's method begins with an initial approximation, x0. The next approximation, x1, is found by doing which of the following?
 A. Linearizing at and finding its root B. Bisecting the line from to C. Approximating with a Taylor polynomial of degree 2 at and finding its root D. Approximating with a Lagrange interpolator and finding its root E. None of the above
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Question 94
Let and . Use Newton's method to find .
Identify the approximation below that is correct to 6 significant digits.
 A. B. C. D. E.
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Question 95
What do we mean when we say Newton's method is not a general purpose method?
 A. It does not always converge. B. It may divide by zero. C. It requires the evaluation of f'. D. It may get stuck in a cycle. E. It requires complex arithmetic.
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Question 96
Which of the following is a Newton iteration to find ?
 A. B. C. D.
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Question 97
Which of the following is a Newton iteration to find ?
 A. B. C. D.
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Question 98
The secant method begins with two approximations, and , to a zero of . The next point, is found by doing which of the following?
 A. Using the secant angle between , and B. Averaging the Newton method and the bisection method C. Approximating with a Taylor polynomial of degree 2 at and finding its root D. Finding the x-intercept of the line joining and E. None of the above
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Question 99
The secant method requires how many function evaluations to complete k iterations?