Question

微积分问题

1. the graph of y=3x^4-16x^3+24x^2+48 is concave down for
A.x<0 B. x>0 C. x< -2 or x> -2/3 D. x<2/3 or x>2 E. 2/3<x<2

2.an equation of the line tangent to the graph of y=cos(2x) at x=π/4 is
A. y-1= -(x-π/4) B. y-1= -2(x-π/4) C. y=2(x-π/4) D. y= -(x-π/4) E. y= -2(x-π/4)

3.let f(x)=x^4+ax^2+b. the graph of f has a relative maximum at(0,1) and an inflectionn point when x=1. the values of a and b are
A. a=1, b= -6 B. a=1, b=6 C. a= -6,b=5 D. a= -6, b=1 E. a=6, b=1

4.a particle moves along a line so that at time t, where 0≤t≤π, its position is given by s(t)= -4cost-(t^2)/2+10. what is the velocity of the particle when its acceleration is 0?

有过程和解答加20分！！！ 拜托，很急！！

Answer 1

1. the graph of y=3x^4-16x^3+24x^2+48 is concave down for
A.x<0 B. x>0 C. x< -2 or x> -2/3 D. x<2/3 or x>2 E. 2/3<x<2
【Solution】:
dy/dx = 12x³ - 48x² + 48x = 12x(x² - 4x + 4) = 12x(x - 2)²
d²y/dx² = 36x² - 96x + 48 = 12(3x² - 8x + 4) = 12(x - 2)(3x - 2)
Let d²y/dx² < 0, ie. 12(3x² - 8x + 4) = 12(x - 2)(3x - 2) < 0
We have 2/3 < x < 2
Ans : E

2.an equation of the line tangent to the graph of y=cos(2x) at x=π/4 is
A. y-1= -(x-π/4) B. y-1= -2(x-π/4) C. y=2(x-π/4) D. y= -(x-π/4) E. y= -2(x-π/4)
【Solution】:
y = cos2x
dy/dx = -2sin2x
When x = π/4, dy/dx = -2sinπ/2 = -2
Let the equation of tangent line be y = -2x + c
When x = π/4, y = cosπ/2 = 0
So, 0 = -2*π/4 + c, c = π/2
y = -2x + π/2
Ans: E

3.let f(x)=x^4+ax^2+b. the graph of f has a relative maximum at(0,1) and an inflectionn point when x=1. the values of a and b are
A. a=1, b= -6 B. a=1, b=6 C. a= -6,b=5 D. a= -6, b=1 E. a=6, b=1
【Solution】:
df/dx = 4x³ + 2ax,
d²f/dx² = 12x² + 2a
When x = 0, y = 1, So, b = 1
When x = 1, d²f/dx² = 0, So, 0 = 12 + 2a, a = -6
Ans: D

4.a particle moves along a line so that at time t, where 0≤t≤π, its position is given by s(t)= -4cost-(t^2)/2+10. what is the velocity of the particle when its acceleration is 0?
【Solution】:
s(t) = -4cost - t²/2 + 10
ds/dt = 4sint - t
d²s/dt² = 4cost - 1
Let d²s/dt² = 0, cost = 1/4, t = 1.318 (s)
So, V = ds/dt = 4sin1.31 - 1.318 = 2.56 (m/s)