Ασκήσεις Ι

ASKHSEIS I1. Na brejeÐ to pedÐo orismoÔ twn parakˆtw sunart sewn: √ 1 − x − 2. √ x+1(1) f1(x) = x2 − 5x + 6, (2) f2(x) = x − 1 , (3) f3(x) =2. Na parastajeÐ grafikˆ h sunˆrthsh 5 x + 7, eˆn − 4 x < −2,  eˆn − 2 x < 3,  2 eˆn 3 x < 5.    f (x) = 2,    x − 1,3. Na melethjeÐ wc proc th monotonÐa kai ta akrìtata h sunˆrthsh f (x) = 3x2 − 1.4. DÐnontai oi sunart seic f (x) = x + 4 kai g(x) = x2 − 9.(1) Na apodeiqjeÐ ìti h sunˆrthsh f eÐnai 1 − 1.(2) Na brejeÐ h antÐstrofh sunˆrthsh f −1 thc f .(3) Na brejeÐ h sunˆrthsh g ◦ f −1 kai na parastajeÐ grafikˆ.5. Na upologisjoÔn ta parakˆtw ìria:(1) lim( √ x + 5x), lim x2 − 1 (3) lim x2 − 3x + 2 lim 3 lim 3 x→4 x→2 (2) x→1 x−1 , x2 − 4 , (4) x→0 x4 , (5) x→0− x7 ,(6) lim ln x, lim 3 (8) lim 1 x (9) lim ef x, (10) lim (2x3 + 5x − 7), x→0+ (7) x7 , x→+∞ 2 x→−∞ x→−∞ , π − 2 x→ lim 2(x − 1)(x2 − 3) (12) lim √ 4x2 − x + 1 − 2x . x→+∞(11) x→−∞ 5x2 + 7 − 2x3 + , 2x6. Na apodeiqjeÐ ìti lim (x − 3) · sun 2013 = 0. x−3 x→37. DÐnetai h sunˆrthsh  eˆn x < 2, αx + 6, eˆn x = 2,  eˆn x > 2.   f (x) = 4α,  x2 + βx,Na prosdioristoÔn oi timèc twn α kai β ètsi ¸ste h f na eÐnai suneq c sto shmeÐo x0 = 2.