题目
设函数f(x）=secx在x=0处的2次泰勒多项式为1+ax+bx^2 ,则（）。A. a=1,b=-1/2B. a=1,b=1/2C. a=0,b=-1/2D. a=0,b=1/2
设函数f(x）=secx在x=0处的2次泰勒多项式为1+ax+bx^2 ,则（）。
- A. a=1,b=-1/2
- B. a=1,b=1/2
- C. a=0,b=-1/2
- D. a=0,b=1/2
题目解答
答案
【解析】D
解析
步骤 1：计算f(x)在x=0处的函数值
f(x) = sec(x) = 1/cos(x)
f(0) = 1/cos(0) = 1
步骤 2：计算f(x)在x=0处的一阶导数值
f'(x) = sec(x)tan(x)
f'(0) = sec(0)tan(0) = 1*0 = 0
步骤 3：计算f(x)在x=0处的二阶导数值
f''(x) = sec(x)tan^2(x) + sec^3(x)
f''(0) = sec(0)tan^2(0) + sec^3(0) = 1*0 + 1^3 = 1
步骤 4：根据泰勒多项式的定义，将f(x)在x=0处的函数值、一阶导数值和二阶导数值代入泰勒多项式
f(x) ≈ f(0) + f'(0)x + f''(0)x^2/2!
f(x) ≈ 1 + 0x + 1x^2/2
f(x) ≈ 1 + 0x + 1/2x^2
f(x) = sec(x) = 1/cos(x)
f(0) = 1/cos(0) = 1
步骤 2：计算f(x)在x=0处的一阶导数值
f'(x) = sec(x)tan(x)
f'(0) = sec(0)tan(0) = 1*0 = 0
步骤 3：计算f(x)在x=0处的二阶导数值
f''(x) = sec(x)tan^2(x) + sec^3(x)
f''(0) = sec(0)tan^2(0) + sec^3(0) = 1*0 + 1^3 = 1
步骤 4：根据泰勒多项式的定义，将f(x)在x=0处的函数值、一阶导数值和二阶导数值代入泰勒多项式
f(x) ≈ f(0) + f'(0)x + f''(0)x^2/2!
f(x) ≈ 1 + 0x + 1x^2/2
f(x) ≈ 1 + 0x + 1/2x^2