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Implement the Trapezoidal Rule for integrating a function. Use the following function:

double trap(double (*pf) (double x), double a, double b, int n)

See Fig. 18-10.

Here, pf points to the function f that is to be integrated, a and b bracket the interval [a, b] over which f is to be integrated, and n is the number of subintervals to use. For example, the call trap ( square, 1,2 , 10 0 ) would return 1.41421. The Trapezoidal Rule returns the sum of the areas of the n trapezoids that would approximate the area under the graph off. For example, if n = 5, then it would return the following, where h = (b-a)/5, the width of each trapezoid.

Fig. 18-10

h 2 Ilf(a) + 2f(a + h) + 2f (a + 2h) + 2f (a + 3h) + 2f (a + 4h) +f (b)]

Fig. 5.12(b)

(a) Ar(1, 2), Br(1, 3), and Cr(4, 3) (b) As( 2, 1), Bs( 3, 1), Cs( 3, 4) (c) A-( 1, 2), B-( 1, 3), C-( 4, 3)

6.2 6.3

18.10 Recognizing rotations Name the rotation that takes ^ABC to (a) ^ArBrCr, (b) ^AsBsCs , and (c) ^A-B-C- as illustrated in Fig. 18-11.

Apply the dereference operator * to the variable * p. The declaration in t & r = n ; declares r to be a reference (alias) for the in t variable n. The assignment p = &n; assigns the address of n to the pointer p.

Fig. 18-11

6.4 The declaration int * q = p ; declares q to be a pointer (memory address) pointing to the same int to which n points. The assignment n = *p ; assigns to n the int to which p points. 6.5 A dangling pointer is a pointer that has not been initialized. It is dangerous because it could be pointing to unallocated memory, or inaccessible memory. fied variable. If a pointer pointing to inaccessible memory is dereferenced, the program will probably crash (i. e., terminate abruptly).

66 If a pointer pointing to unallocated memory is dereferenced, it could change the value of some unidenti.

(a) 270 clockwise or 90 counter-clockwise about the origin, P(x, y) A Pr( y, x) (b) 180 about the origin (either clockwise or counter-clockwise), P(x, y) A Ps( x, y) (c) 45 clockwise, P(x, y) A P-(x cos 45 P-a 22 x 2 22 22 y, y 2 2 22 xb 2 y sin 45, y cos 45 x sin 45 )

6.8 6.9

18.11 Naming rotations Name the transformation that rotates clockwise about the origin: (a) 20 (b) 30 (c) 60 (d) 75

You cannot have a reference to a constant; it s address is not accessible. The reference operator SC cannot be applied to a constant.

Now suppose we click on Continents. The drop-down menu appears as in Fig. 5.13(a), with a check mark in front of Africa. If we then click on Africa, the corresponding area will be displayed, as shown in Fig. 5.13(b). Now if we again click on Continents, the drop-down menu will again appear but now the check mark preceding Africa will not appear, having been toggled off. If this cycle (Continents/Africa) is repeated, the check mark will alternately reappear, then disappear, etc.

(a) P(x, y) A Pr(x cos 20 (b) P(x, y) A Pr(x cos 30 Pr(0.866x y sin 20 , y cos 20 y sin 30 , y cos 30 0.5x) x sin 20 ) x sin 30 ) Pr(0.9397x Pr a 23 x 2 0.3420y, 0.9397y 1 23 y, y 2 2 1 xb 2 0.3420x)

6.10 The variable p has type char, while the expression &c has type pointer to char. To initialize p to 6~ c, p would have to be declared as type char * . 6.11 Static binding is when memory is allocated at compile time, as with the array declaration:

x sin 60 )

double a[400];

Dynamic binding is when memory is allocated at run time, by means of the new operator:

x sin 75 )

double* p; P = new double[400];

0.9659x)