课程名称︰微积分甲 课程性质︰必修 课程教师︰陈其诚 开课学院：理学院,管理学院 开课系所︰生工,生机,地质,工业管理组 考试日期（年月日）︰2009.10.23 考试时限（分钟）：100分钟 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : ◎ x^2 = x的平方, _0_ = 下标 Write down your answers on the answer sheet. You should include all the neces- sary calcultions and reasoning. This exam contains seven proplems written in one page. (1) Calculate the derivatives of the following functions(5 point each) a.f(x)=x^10+23x+8 b.f(x)=(x^2+x+1)^-1 c.f(x)=(1+sinx)[(1+cosx)^-1] d.f(x)=g(sinx), with g'(t)=(1-t^2)^1/2 e.f(x)=(cosx*cosx+1)^1/9 (2) Calculate the following limits(5 points each) a.lim_x→0_sin9x/x b.lim_x→+∞_(x^2-4)/(x^4+1)^1/2 (3) (10 points) Find the tangent x_0_=-0.7, Calculate x_1_ (4) (10 points) Find the tanget line to the curve x^7+7xy-7y^7=1 at (1,1). (5) Let f(x)=x(x-10)^3 a.Determine the intervals where f is increasing or decreasing(6 points). b.Determine the intervals wherer f is concave up or concave down(6 points). c.Find all points of inflection(3 points.) d.Sketch the graph y=f(x) (7 points). e.Does f(x) have a maximum or minimum for x in(+∞,∞)? Why(3 points)? (6) (10 points)You are in a boat that is 10km off the shore. Suppose you are heading for a place located 15km down the coast and 5km inland and you plan to take your motor bike to the place after you get off the boat. The gasoline consumption is 0.2 L/km for the boat, 0.1 L/km for the motor bike.Use your computation to show that to save the most amount of the gasoline, you will need to get off the boat 5km down the cost. (7) a.(5 points) Suppose f"(x) exists and is positive for every x in R, and suppose we know that f(0)=0, f'(0)=3. Can we conclude that f(x)>3x for x>0? Why? b.(5 points) Give an example of a degree 4 polynomial that has no point of inflection on R --

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