Pre-Calc Homework Solutions 77

Pre-Calc Homework Solutions 77 - Section 3.3 1 x 77 31. (a)...

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Unformatted text preview: Section 3.3 1 x 77 31. (a) Note that x0 x 1 x sin x, for all x, so lim x sin 0 by the Sandwich Theorem. 0. sin 1 h 1 oscillates h 9. These are all constant functions, so the graph of each function is a horizontal line and the derivative of each function is 0. x h x Therefore, f is continuous at x (b) f (0 h) h f (0) 1 h sin h 10. (a) f (x) lim h0 lim h0 f (x h) h f (x) 1 lim h0 h 0 x h x h lim h0 1 h (c) The limit does not exist because sin between (b) f (x) lim h0 lim h0 1 and 1 an infinite number of times 0 (that is, for h in any open h) h x (x h) hx(x h) x(x h) f (x f (x) lim h0 x h x h h h) 2 h0 hx(x lim arbitrarily close to h interval containing 0). lim h0 x2 x Section 3.3 Exercises 1 h (d) No, because the limit in part (c) does not exist. (e) g(0 h) h g(0) h 2 sin 0 1. h h sin 1 h dy dx d 2y dx 2 dy dx d 2y dx 2 dy dx d 2y dx 2 dy dx d 2y dx 2 dy dx d ( x 2) dx d ( 2x) dx d 1 3 x dx 3 d 2 (x ) dx d (2x) dx d (2) dx d 2 (x ) dx d (2x) dx d 1 3 x dx 3 d (3) dx 2x 0 2x 2 d (x) dx d (1) dx d (1) dx As noted in part (a), the limit of this as x approaches zero is 0, so g (0) 0. 2. x2 2x 2 0 1 0 2 2x s Section 3.3 Rules for Differentiation (pp. 112121) Quick Review 3.3 1. (x 2 x 2. x x2 1 2 x 2x 3 2x 2 1 3 3. 0 d d (x) (1) 2x dx dx d (1) 2 0 2 dx d 1 2 x dx 2 d (x) dx d (x) dx d (x) dx d (1) dx d 2 (x ) dx 2)(x 1 1) x 2 2x 1 1 x 2x 2 1 x2 x 1 x2 1 1 x 1 2x 1 2 1 4. 1 0 2x 1 x2 x 5 x2 4 2 1 3 x 5x 2 x 1 5. 3. 3x 2 4. 3x 4 3x 2 3x 4 2x 2 2x 2x 3 2x 2 x2 x 2x 2x 2 2 x 1 2x d 3 (x ) dx 4 2x 2 3 2 x 2 d y dx 2 2 d 2 (x ) dx d (1) dx 1 0 2x 1 5. (x x 1 x 6. 7. x 2)(x x x 2 1) 2x 1 x 2 1 x 2 x 1 1 2 1 6. 2 x 2 dy dx 0 x (x ) x 2 1 2x 3x 2 d (2x) dx 1 2x 3x 2 3 x d 2y dx 2 d ( 1) dx d (3x 2) dx 0 7. [0, 5] by [ 6, 6] 2 6x 2 6x d (2x 2) dx d (15) dx dy dx d 2y dx 2 d 4 (x ) dx d (7x 3) dx 4x 3 At x 1.173, 500x 6 1305. At x 2.394, 500x 6 94,212 After rounding, we have: At x 1, 500x 6 1305. At x 2, 500x 6 94,212. 8. (a) f (10) (b) f (0) (c) f (x h) 7 7 7 f (a) a 7 lim xa x 7 a 21x 2 d (4x 3) dx 4x 0 4x 3 21x 2 4x 12x 2 42x 4 d (21x 2) dx d (4x) dx 8. dy dx d 2y dx 2 dy dx d y dx 2 2 d d (5x 3) (3x 5) 15x 2 15x 4 dx dx d d (15x 2) (15x 4) 30x 60x 3 dx dx d (4x 2) dx d (8x) dx d (1) dx 3 9. 8x lim 0 xa 3 8 0 8x d (8) dx 8 24x 4 f (x) (d) lim x xa 0 d ( 8x 3) dx 0 24x 4 ...
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This note was uploaded on 10/05/2011 for the course MAC 1147 taught by Professor German during the Fall '08 term at University of Florida.

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