that returns a function: the derivative of a function f. Assuming that f is a single- variable mathematical function, its derivative will also be a single-variable function. When called with a number a, the derivative will estimate the slope of f at point (a, f(a)). Recall that the formula for finding the derivative of f at point a is: where h approaches 0. We will approximate the derivative by choosing a very small value for h. The closer h is to 0, the better the estimate of the derivative will be. def make_derivative (f): """Returns a function that approximates the derivative of f. >>> def square (x): f'(a) = lim h→0 Recall that f'(a) = (f(a+h)-f(a))/h as h approaches 0. We will approximate the derivative by choosing a very small value for h. ... ... f(a+h)-f(a) h #equivalent to: square = lambda x: X*X return x*x >>> derivative = make_derivative (square) >>> result = derivative (3) >>> round (result, 3) # approximately 2*3 6.0 Hu h=0.00001

that returns a function: the derivative of a function f. Assuming that f is a single- variable mathematical function, its derivative will also be a single-variable function. When called with a number a, the derivative will estimate the slope of f at point (a, f(a)). Recall that the formula for finding the derivative of f at point a is: where h approaches 0. We will approximate the derivative by choosing a very small value for h. The closer h is to 0, the better the estimate of the derivative will be. def make_derivative (f): """Returns a function that approximates the derivative of f. >>> def square (x): f'(a) = lim h→0 Recall that f'(a) = (f(a+h)-f(a))/h as h approaches 0. We will approximate the derivative by choosing a very small value for h. ... ... f(a+h)-f(a) h #equivalent to: square = lambda x: XX return xx >>> derivative = make_derivative (square) >>> result = derivative (3) >>> round (result, 3) # approximately 2*3 6.0 Hu h=0.00001

C++ Programming: From Problem Analysis to Program Design

8th Edition

ISBN:9781337102087

Author:D. S. Malik

Publisher:D. S. Malik

Chapter13: Overloading And Templates

Section: Chapter Questions

Problem 30SA

See similar textbooks

Similar questions

) Consider the following functions. Decide whether these functions are injective,surjective, and invertible. Justify your answer (e.g., if you claim that a function is invertible, you need togive a justification as to why you think that function is invertible). Give counterexamples when needed.You can draw arrow diagrams to help justifying your answer.a) Function f: ℤ × ℤ → ℤ is defined as f((a, b)) = 2b – 4a.b) A = {1, 2, 3}. Function f: ?(A) → {0, 1, 2, 3} is defined as f(X) = |X| where |X| = size of X. Forexample, |{1, 2}| = 2. ?(A) is the power set of A.c) Function f: {0, 1}3 → {0, 1}3 is defined by the following rule. For each string s ∈ {0, 1}3,f(s) = f(x1x2x3) = x3x1x2, where x1, x2, x3 ∈ {0, 1}. For example, if x1 = a, x2 = b, and x3 = c, thenf(abc) = cab. Another example: f(011) = 101.
Consider the following functions. Decide whether these functions are injective, surjective, and invertible. Justify your answer (e.g., if you claim that a function is invertible, you need to give a justification as to why you think that function is invertible). Give counterexamples when needed.
Write a Racket function "combine" that takes two functions, f and g, as parameters and evaluates to a new function. Both f and g will be functions that take one parameter and evaluate to some result. The returned function should be the composition of the two functions with f applied first and g applied to f's result. For example (combine add1 sub1) should evaluate to a function equivalent to (define (h x) (sub1 (add1 x))). You will need to use a lambda function to achieve your result. Essentially you want (combine f g) to evaluate to a lambda function with f and g embedded inside. You can test your combine with invocations such as ((combine add1 add1) 1), which should evaluate to 3 since the anonymous function returned by combine is being applied to 1. Note: This is an example of a "function closure". f and g are defined in the scope of combine, but the lambda embeds a copy of them in the lambda function that can be used outside of combine's scope. When creating a lambda function in…
A triple (x, y, z) of positive integers is pythagorean if x2 + y2 = z2. Using the functions studied in class, define a function pyth which returns the list of all pythagorean triples whose components are at most a given limit. For example, function call pyth(10) should return [(3, 4, 5), (4, 3, 5), (6, 8, 10), (8, 6, 10)]. [Hint: One way to do this is to construct a list of all triples (use unfold to create a list of integers, and then a for-comprehension to create a list of all triples), and then select the pythagorean ones. def unfold[A, S](z: S)(f: S => Option[(A, S)]): LazyList[A] = f(z) match { case Some((h, s)) => h #:: unfold(s)(f) case None => LazyList() } code in scala
Prove or disprove:(a)∃ sets A,B ∀ functions f: A→B, f is a bijection. (b) if f : A → B is a function, C ⊆ A, and D ⊆ B, then f : C → D is always a function.[Note 1: one is true, the other is false.][Note 2: feel free to draw bubble-arrow pictures here, but make sure you explain what’s going on very carefully.]
Q. Taylor Series expansion of sine for x is computed as follows: sin x=x - x^3/3! + x^5/5! - x^7/7!+... Let the value of x be 30, hence radian value for 30 degree is 0.52359 sin(0.52359) = 0.52359 - 0.52359^3/3! + 0.52359^5/5! - 0.52359^7/7!+ ... (a)Write a function in c++ that takes value of x in degrees and returns the value of x in radians. (b)Write a function in c++ that calculates and returns the factorial of a number.e.g. 3!=6 (c)Write a function in c++ that calculates and returns power of a number. e.g. 23=8. (d)Use the above three functions to find the taylor expansion of sine for in degrees as given above.(Calculate first 3 terms)
If p, q are positive integers, fis a function defined for positive numbers and attains only positive values such that f(xf(y)) = x ^ p * y ^ q then prove that p ^ 2 = q
explain how the following function works step by stepFunction: computeFa Parameters: a - Dimension a of the scrapper num steps number of steps for determining h Description: Computes the value of TA(a) for the dimension ac by applying the Trapezoidal rule for integrating f(x) from a to ta double computeFa (double a, int num_steps) int i=0; double integral = 0; double interval = (a+a)/num steps: double XL = -a; // left hand side of trapezoid double XR = -a interval; // right hand side of trapezoid while (i<num_steps) integral = integral + ((pow (a, 2) - pow((-XL), 2)) pow (M_E, -((XL) /a)) + (pow (a, 2) pow((-XL), 2)) pow (M_E, -((XL)/a))) interval/2; XL = XR; XR = XR + interval; return (integral);
For each of the following functions, indicate the class (g(n)) the function belongs to. (Use the simplest g(n) possible in your answers.) Please show work to prove the assertions.
(6a) how many different functions f : A → B can be defined if m=3 and n=5? (6b) if m=4 and n=6 then how many different one-to-one functions f : A →B can be defined? (6c) if m=5 and n=4 then how many different onto functions f : A → B can be defined?
Define a function print_variables_over_time(c_0, tn_0, tp_0, percent_reduction, final_year) that accepts current levels for CHLA, TN, and TP and prints out a table showing a forecast of these values over time, provided all are reduced by the given percentage each year. Parameter summary c_0, tn_0, and tp_0 are initial CHLA, TN, and TP values respectively (i.e. at year 0). percent_reduction is the percentage reduction in all three values, per year, and will be given as a value between 0 and 100. final_year = number of the last year to print, e.g. 4 means print years 0, 1, 2, 3, 4. You can assume final_year is at least 1. Notes: Use 'Year CHLA TN TP' for the header row. The year should be printed with width 3 (ie, :3d will be helpful) CHLA, TN, and TP should be printed with width 10 and 2 decimal places. (Hint: :10.2f will help.) You must use a while loop for this - it will help with the next question! You cannot use any for loops or list comprehensions.
Which functions are one-to-one? Which functions are onto? Describe the inversefunction for any bijective function.(a) f : Z → N where f is defined by f (x) = x4 + 1(b) f : N → N where f is defined by f (x) = { x/2 if x is even, x + 1 if x is odd}(c) f : N → N where f is defined by f (x) = { x + 1 if x is even, x − 1 if x is odd}