function f(t) 1 tn eat sin at cos at sinh at cosh at e-at f(t) U(ta) or Ua(t) (a ≥ 0) (ta) (a> 0) U(t − a)f(t − a) or Ua(t)f(t − a) f(n)(t) (−t)" f (t) - (fg)(t) = f(r)g(t − T) dr Laplace transform F(s) 1/s (s > 0) n!/sn+1 (s > 0) 1/(s – a) (s> a) a/(s²+a²) (s > 0) s/(s² + a²) (s > 0) a/(s² - a²) (s> |a|) s/(s² – a²) F(s+a (s>|a|) -as e S S 0) as e as e S F(s F(n)(s) F(s)G(s) sn ¯¹ƒ (0) — sn−² ƒ'(0) … … … — ƒ (n−¹) (0) Solve the differential equation บ'' "-y-12y=-4-5₁(t), y(0) = −2, y'(0) = 1 using Laplace transforms. The solution is y(t) and y(t): = = fort > 4 for 0 < t <4

Let f(x)= = 6 sin x 6 sin + 4 cos a Then f'(x) = π The equation of the tangent line to y = f(x) at a = can be written in the form y = mx + b where 2 -m= and b=

5 i+ cos' (61)j + lIn (2r³)k. t – 2 Calculate the derivative of the function r(t) = -

1. Find f. f ''(?) = sin(?) + cos(?),    f(0) = 1,    f '(0) = 2 f(?) =

i) Let f(r) = e2-g(x) where g(2) = -1 and g'(2) = 7, and g"(2) to calculate f"(2). Show your work. 3. = -4. Use the product rule twice Hint. e2 = (e")2. ii) Let F(x) = f(xf(rf(x))) where f(1) = 2, f(2) = 3, f'(1) = 4, f'(2) = 5, and f'(3) = 6. Calculate F'(1). Show your work.