Test 2 practice problems-solutions

Test 2 practice problems-solutions - Practice Problems for...

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Unformatted text preview: Practice Problems for Test 2 Answers 1) Use the definition: f (x) = lim h 0 f (x + h) - f (x) to show that h f (x) = 4x - 3 if f (x) = 2x 2 - 3x f (x) = 2x 2 - 3x f (x + h) = 2(x + h) 2 - 3(x + h) = 2(x 2 + 2xh + h 2 ) - 3(x + h) f (x + h) - f (x) f (x) = lim h 0 h 2 (2x + 4 xh + 2h 2 - 3x - 3h) - (2x 2 - 3x) f (x) = lim h 0 h h(4x + 2h - 3) lim = 4 x + 2(0) - 3 = 4x - 3 h0 h x + 1 if x 2 2) Use g(x) = 2x -1 if x > 2 a) Is g(x) continuous for all x? Each "piece" is continuous so we just need to check where they meet: x 2 lim- x + 1 2 + 1 = 3 = x 2 + lim 2x -1 = 2(2) -1 = 3 g(2) = 2 + 1 = 3 Therefore g(x) is continuous. b) Is g(x) differentiable at for all x? Each "piece" is differentiable so we just need to check where they meet: 1 if x < 2 g(x) = 2 if x > 2 Therefore g(x) is NOT differentiable at x = 2 . g- (2) g (2) . + c) Sketch the graphs of g(x) and g (x) . 3) Find a) dy = -9t5 cos2 (t 3 )sin(t 3 ) + 3t 2 cos3 (t 3 ) dt dy d) = (t 2 + 5)9 (-21t 2 + 20t - 5) dt dy 2 f) = t [t sec 2 (t - 1) + 3tan(t - 1)] dt dy - sint i) = dt 1 + cos2 t dy 1 l) = dt cos y 1 - t 2 4) 5) a) y = -5(x - p ) dy for each of the following: dt dy 1 - 5sin(5t) cos(5t) = dt -3cos(3y)sin(3y) dy e) = 2p 2t cos(p 2t 2 ) dt dy g) = (-sin t)(cos(cos t)) dt 5t t t dy e (5(1+ 3 ) - ln 3(3 )) = j) dt (1+ 3t ) 2 b) c) dy 5 = dt 10t - 5 dy =0 dt 3 dy k) = (ln5)(12t 2 )5 4 t dx h) b) y = 8 - 9x 5 c) y = - 1 x + 32 8 dx dx dr = = -2r sin(q ) = -2cos(q) sin(q ) dq dr dq (b) k(t) = z(y(t)) k (t) = z(y(t)) y (t) k (3) = z(y(3)) y (3) = z(2)(5) = (-3)(5) = -15 6) (a) h(t) = z(t) y(t) h (t) = z(t) y (t) + y(t) z(t) h (3) = z(3) y (3) + y(3) z(3) = (-1)(5) + (2)(-2) = -5 - 4 = -9 7) dy = 2xsin x(x cos x + sin x) dx b) 100 d2 y 2 2 2 2 2 2 = 2sin x + 8x sin x cos x + 2x cos x - 2x sin x dx 8) a) 80 , 120 9) 10) ...
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