1、微生一微積分 期末考 2009/1/5 科目代螞：BMB11201參考解答：1. 30% Differentiate the functions with respect to the independent variable . (a) (b) (c) (d) (e) (f) 解：(a) (b) (c) (d) (e) (f) 2. 14% (a) Find the tangent line at x=1 of f(x)= (b) Assume that x is a function of t. Find dy/dt when y=cos(x) and dx/dt= 解：(a) , at

2、x=1 f '(x)=f '(1)= -1/2 the tangent line : y-f(1)=f'(1)(x-1) or (b) 3. 24% (a) Suppose that f '(x)=3x-2, find at x=2 (b) For xy-y3=1 , find dy/dx at x=0 . (c) Use the tangent line approximation to estimate e0.2 . 解：(a) (b) , at x=0 , y= -1 , get y' = (c) conider f(x)=ex, at x=0 t

3、he tangent line : y-f(0)=f'(0)(x-0) or y-1=f'(0)(x)=x or y=x+1 , hence e0.2=f(0.2)0.2+1=1.2 , 4. 12% For the Monod growth function a, k are positive constants. (a) Differentiate f (R) (b) Describe the following graph about f (R) and f '(R) . 解：(a) (b) f '(R)>0 so the function f (R

4、) is increasing. however f '(R) is decreasing, this means that f (R) reduces the increase when R becomes larger. 5. 14% For f(x)=3x+e2x , (a) Use the equalityto find (b) Find 解：(a) (b) f'(x)=3+2e2x and f(0)=1 ,f-1(1)=0 6. 16% Suppose that we use linear approximation to estimate the percentage error of f(d) from a measurement of d, where f(d)=ln d . (a) At d=10 the percentage error for d =5%. Find the percentage error of f(d) . ( ln102.3026 )(b) How accurately must we measure d if we want to es