课程名称︰微积分甲上
课程性质︰必修
课程教师︰周青松
开课学院：理学院
开课系所︰地质科学系 生物环境系统工程系 生物产业机电工程系 工管系科管组
考试日期（年月日）︰104/1/12
考试时限（分钟）：110分钟
试题 :
(1) (10 pts)
Let F(x) = ∫x t(t-3)^2 dt
0
(a) Find the critical points of F and determine the intervals on which F
increases and the intervals on which F decreases.
(b) Determine the concavity of the graph of F and find the points of
inflection.
(c) Sketch the graph of F
(2) (10 pts)
Let f be everywhere continuous and set t F(x) =∫x [t∫t f(u)du] dt
0 1
Find F''(1)
(3) (10 pts)
Find ∫(x sin^2(x) + x^2 sinxcosx ) dx
(4) (10 pts)
Derive ∫cos^2(x)dx = x/2 + (sin2x)/4 + C
(5) (10 pts)
Calculate ∫1/ √[a^2 – (x+b)^2] dx where a > 0
(6) (10 pts)
Calculate∫1/ √[a^2 + (x+b)^2] dx where a > 0
(7) (10 pts)
Show that d/dx sinh^(-1) x = 1/[√x^2+1]
(8) (10 pts)
Calculate ∫1/√(a^2 + x^2) dx = sinh^(-1) (x/a) + C where a > 0
(9) (10 pts)
Show that∫sin^n x dx = -1/n sin ^(n-1) xcosx + (n-1)/n∫sin^(n-2) x dx
(10) (10 pts)
Show that∫π/2 sin^n x dx = (n-1)/n∫π/2 sin^(n-2) x dx
0 0