how to do the questions that I circled?

TECHNIQUES FOR INTEGRATION (Chapter 16) 389 curve has gradient function dy - = 1 - ex and passes through (3, es). Find the equation of the Curve 4 Find f(x) given that: a f'(x) = x2 - 4cosx and f(0) = 3 b f'(x) = 2cosx - 3sina and f (1) = c f'(x) = Vx - 2sinx and f(0) = -2 d f'(x) = ex + 3cosx and f(7) = 0. A curve has gradient function f'(x) = ax + 1 where a is a constant. Find f(x) given that f(0) = 3 and f(3) = -3. 6 A curve has gradient function f'(x) = ax2 + ba where a, b are constants. Find f(x) given that f(-1) = -2, f(0) = 1, and f(1) = 4. Example 6 Self Tutor Find f(x) given that f"(x) = 12x2 - 4, f'(0) = -1, and f(1) = 4. If f"(x) = 12x2 -4 : f'(x) = (12x2 - 4) dx 12x3 - 4x + c 3 = 4x3- 4x + c But f'(0) = -1, so c=-1 Thus f'(x) = 4x3 - 4x - 1 :. f(x) = (4x3 - 4x - 1) dx 4x4 4x2 4 2 -x+d =x4- 2x2 - x +d But f(1) = 4, so 1 -2 - 1 + d =4 and hence d =6 Thus f(x) = 24 -2x2 - x+6 7 Find f(x) given that: a f"(x) = 2x + 1, f'(1) =3, and f(2) = 7 b f"(x) = 15\\x+ , f'(1) = 12, and f(0) = 5 c f"(x) = cosx, f' (#) =0, and f(0) = 3 d f"(x) = 2x and the points (1, 0) and (0, 5) lie on the curve y = f(x). Suppose f"(x) = 3e-x, f(1) = 3, and f(3) = -2. Find f(x)