Calculus: Analytic Geometry and Calculus, with Vectors

426 Curves, lengths, and curvatures points all travel approximately the same distance 2axk and that the segment generates a surf ...

7.2 Lengths and integrals 427 The right member of (2) is an example of a curve integral, and it is denoted by the symbol in (3) ...

¢28 Curves, lengths, and curvatures be a stupid waste of time if it were not for the fact the symbols are useful. They appear ev ...

7.3 Center and radius of curvature 429 most elementaryof elementary geometry tells us that the points obtained for different val ...

430 Curves, lengths, and curvatures A similar procedure, in which the multipliers -x'(t + At) and x'(t) are used, gives the seco ...

7.3 Center and radius of curvature 431 x and ybe functions oft having two derivatives each over some interval and let C be the c ...

432 Curves, lengths, and curvatures Since we want a formula for 0'(t), it is reasonable to differentiate (7.385) with respect to ...

7.3 Center and radius of curvature 433 Since do/ds = b'(t)/s'(t) = ¢'(t)/jv(t)I, the formulas (7.387) and (7.3852) yield the for ...

434 Curves, lengths, and curvatures and Y depend upon t. Supposing P moves along the graph of y = kx2 in such a way that x = t a ...

7.3 Center and radius of curvature 435 10 As an alternative to (or in addition to) Problem 8, find the radius of curva- ture of ...

436 shows that the approximate formula (2) 2R Curves, lengths, and curvatures gives good results when x is small in comparison t ...

7.3 Center and radius of curvature 437 where n is the unit normal obtained by rotating t clockwise through the angle a/2, ds/dt ...

8 Trigonometric Trigonometric functions 8.1 Trigonometric functions and their derivatives We are study- ing and perhaps even lea ...

8.1 Trigonometric functions and their derivatives 439 lengthand provetohim that the arc IV Q(h a 2 _h) has alengthwhichwe may de ...

440 Trigonometric functions (8.12), and the number 0 is negative. Even though we could pretend to be appalled by the idea that n ...

8.1 Trigonometric functions and their derivatives 441 Since 1 - cos 0 > 0 and 1 + cos 0 > 1 and sin 8 < 0, the last ter ...

442 Trigonometric functions These formulas, which we have used many times, have at long last been proved. For values of x for wh ...

8.1 Trigonometric functions and their derivatives 1.0 0.8 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.6 0.4 0.5 0.4 0.3 02 0.1 0 0 ...

444 Trigonometric functions y = sin x to obtain a procedure for sketching the graph of y = 3 sin x, and sketch the graph. 3 Equa ...

8.1 Trigonometric functions and their derivatives 445 9 Find the length L of the shortest ladder thatcan A reach from level grou ...