曉文 發問時間: 科學數學 · 6 年前

大學數學微積分~~~急 20點

1.The length l of a rectangle is decreasing at the rate of 2cm/sec while the width w is increasing at the rate of 2cm/sec .When l=12cm and w=5cm ,find the rates of change of (a) the area (b) the perimeter (c) the lengths of the diagonals of the rectangle

2.A 5-m ladder is leaning against a house when its base stars to slide away .By the time the base is 4 m from the house,the base is moving at the rate of 3m/sec

(a)At what rate is the area of the triangle formed by the ladder,wall,and ground changing then

(b)At what rate is the angle θ between the ladder and the ground changing then

3.A girl flies a kite at a height of 90m,the wind carrying the kite horizontally away from her at a rate of 8m/sec.How fast must she let out the string when the kite is 150 m away from her

4.A light shines from the top of a pole 15m high .A ball is dropping from the same height from a point 9 m away from the light .How fast is the shadow of the ball moving along the ground 1/2 sec later (Assume the ball falls a distance s=4.9t^2 m in t sec)

5.A spherical iron ball 20 cm in diameter is coated with a layer of ice of uniform thickness .If the ice melts at the rate of 160cm^3/min,how fast is the thickness of the ice decreasing when it is 5 cm thick? How fast is the outer surface area of ice decreasing

6.A highway patrol plane flies 5 km above a level,straight road at a steady 190km/hr.The pilot sees an oncoming car and with radar determines that at the instant the line-of-sight distance from plane to car is 8 km ,the line-of-sight distance is decreasing at the rate of 260km/hr.Find the car's speed along the highway

7.Suppose that f(-1)=3 and that f '(x)=0 for all x.Must f(x)=3 for all x? Give reasons for you answer

8.Assume that f is differentiable on a≤x≤b and that f(b)<f(a).Show that f ' is negative at some point between a and b

1 個解答

評分
  • 麻辣
    Lv 7
    6 年前
    最佳解答

    1.(a) A'=?A=w*LA'=d(w*L)/dt=w*L'+L*w'=-2w+2L=2(L-w)=2(12-5)=14 cm^2

    (b) the perimeter P'=?P=2(w+L)P'=2d(w+L)/dt=2(w'+L')=2(2-2)=0

    (c) the lengths of the diagonals of the rectangle G'=?G=2√(w^2+L^2)G'=2d√(w^2+L^2)/dt=2*0.5*(2ww'+2LL')/√(w^2+L^2)=2(ww'+LL')/√(w^2+L^2)=2(5*2-12*2)/√(25+144)=-28/13

    2. (a) a^2+b^2=252(aa'+bb')=0a'=-bb'/a=-4*3/3=-4 m/s

    A=a*b/2A'=d(ab)/2dt=(ab'+ba')/2=(3*3-4*4)/2=-7/2 m^2/s

    (b) θ=atan(a/b) => tan(θ)=a/b(secθ)^2*θ'=(ba'-ab')/b^2θ'=(ba'-ab')(cosθ)^2/b^2=(-4*4-3*3)(4/5)^2/16=-25*16/25*16=-1 rad/s

    3. L^2=x^2+h^2 => LL'=xx'L'=xx'/L=√(L^2-h^2)x'/L=√(150^2-90^2)8/150=6.4 m/s

    4. x=Shadowy=4.9t^2 => y'=9.8ttanQ=y/9=(12-y)/x => x=(108-9y)/yx'=-[9y+(108-9y)]y'/y^2=-108y'/y^2=-108*9.8t/4.9^2*t^2=-216/4.9t=-216/4.9*0.5=-88.16 m/sminus=shorten

    5. V1=4πr^3/3V2=4π(r+T)^2/3V=V2-V1=[(r+T)^3-r^3]4π/3V'=(r+T)^2*T'4πT'=V'/4π(r+T)^2=160/4π(10+5)^2=40/225π=0.1778 cm^3/min

    A=4π[(r+T)^2-r^2]A'=8π(r+T)T'=8π(10+5)*40/225π=8*15*40/15^2=64/3=21.33 cm^2/min

    6. x=car runs along the highwayx^2=L^2-h^2 => xx'=LL'x'=LL'/x-190=LL'/√(L^2-h^2)-190=8*260/√(64-25) - 190=143 kph

    7. Give reasons for you answerf'(x)=0 => ∫f'(x)dx =∫0dx + c=> f(x)= 0 + c = cBoudary condition is f(-1)=c=3Thus f(x)=3 for all x

    2014-11-15 18:15:44 補充:

    8.Assume that f is differentiable on a≤x≤b and that f(b)

    y'(x)=e^x

    slope=y'(x) >= 0 forever

    Thus it can't exist!!

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