Concept explainers
Looking ahead (to Chapter 9) Suppose that a function f has derivatives of all orders near x = 0. By the Fundamental Theorem of Calculus,
- a. Evaluate the integral using
integration by parts to show that
- b. Show that integrating by parts n times gives
This expression, called the Taylor series for f at x = 0, is revisited in Chapter 9.
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CODE/CALC ET 3-HOLE
- 2. Consider the Karnaugh map of a Boolean function k(w, x, y, z) shown at right. I (a) Use the Karnaugh map to find the DNF for k(w, x, y, z). (b) Use the Karnaugh map algorithm to find the minimal expression for k(w, x, y, z). x y z h(x, y, z) 0 0 1111OOOO: 0 0 0 0 нноонно 10 1 1 LOLOLOL 3. Use a don't care Karnaugh map to find a minimal representation for a Boolean expression h(x,y,z) agreeing with the incomplete I/O table below: 1 0 0 0 1 OLO 0 0 NE IN xy yz 1 IN WX yz 1 ÿz 1 wx wx wox xy xy fy 1 1 1 1arrow_forwardGiven the following function: f(x) = 2x For g(x) = Sf(x) dx, determine g(x).arrow_forwardSimplify the following Boolean functions, using Karnaugh map:F(A,B,C,D) = ∑(0, 2, 4, 5, 6, 7, 8, 10, 13, 15)arrow_forward
- Express the Boolean function F = y + x z as a sum of minterms. Find the following: (a) List the -minterms of the function. (b) List the -minterms of F' (c) Express T.T in max of minterms in algebraic form. (d) draw the max term function.arrow_forwardCreate K-maps and then simplify for the following functions 1) F(x, y, z) = xyz’ + x’yz + xyzarrow_forwardGiven the Boolean function F(x,y,z) = Σ(0,6), simplify it using the Karnaugh map. Be sure to type/write your complete solution.arrow_forward
- Using Matlab, find the positive minimum point of the function f(x) = x^-2 * tan(x) by computing the zeros of f' (derivative of f) using Secant's methodarrow_forward2 Simplify the following Boolean function, using four variables K-map. F(A, B, C, D) = E(2, 3, 6, 7, 12, 13, 14)arrow_forwardReduce this term into normal from, if it exists (show your working):(λx.((λz.zwz)(((λxyx.y)(λx.y)(λy.x))((λx.xx)(λy.yyy)))))tarrow_forward
- The Boolean function F(A,B, C,D) ={m (1,2, 3, 5, 6, 7, 13,15)| Use K-map to simplify the equationarrow_forward2. calculates the trajectory r(t) and stores the coordinates for time steps At as a nested list trajectory that contains [[xe, ye, ze], [x1, y1, z1], [x2, y2, z2], ...]. Start from time t = 0 and use a time step At = 0.01; the last data point in the trajectory should be the time when the oscillator "hits the ground", i.e., when z(t) ≤ 0; 3. stores the time for hitting the ground (i.e., the first time t when z(t) ≤ 0) in the variable t_contact and the corresponding positions in the variables x_contact, y_contact, and z_contact. Print t_contact = 1.430 X_contact = 0.755 y contact = -0.380 z_contact = (Output floating point numbers with 3 decimals using format (), e.g., "t_contact = {:.3f}" .format(t_contact).) The partial example output above is for ze = 10. 4. calculates the average x- and y-coordinates 1 y = Yi N where the x, y, are the x(t), y(t) in the trajectory and N is the number of data points that you calculated. Store the result as a list in the variable center = [x_avg, y_avg]…arrow_forward4. Represent the given Boolean function geometrically, using Carnaugh map and Unit cube, and minimize it. f = y² + (x → ÿ).arrow_forward
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