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• 6. (a) Give the defnition of the directional derivative of f(x, y)at (a,b) in the direction of a uni
• 2.=?
• 7.y=tanx,與x軸所夾區域繞x-軸旋轉的旋轉體體積=________
• 7. Let f :[0,3]→[0,3] be a continuous function with f(0)=0 andf(3)=3. If f is one-to-one and , then=
• 【題組】(b)Hint: for a ＞ 0,a ≠ 1.
• 28. 設a，b為實數，若方程式x4−5x3＋2x2＋ax＋b＝0有一虛根 ，則a＋b＝(A)−10(B)−12(C)−14(D)−16
• (b) Show that the directional derivative of the following func-tion exists in every direction u = (u
• 1. 請問0.1奈米等於多少公尺？(A) 10n (B) 10p (C) 100p (D) 100μ
• 3.(7分)Evaluate the improper integral by transforming it into adefinite integral of the form via an
• 8.=______
• 7. (10 points) Determine and classify the stationary points of the function f(x,y)=xy(3x+6y-2).
• 8. (10 points) Find the total arc length of the cardioid r = a(1-cosθ),a＞0.
• (E). Let g(x) be the inverse function of the strictly increasing function f(x) =x5 + 3x3 + 1. Theng(
• (c) Show that the above function f(x, y) is not differentiableat (0,0)
• 8. The radius of convergence of the power series is___________.
• 8. Find the following limits.【題組】(a)
• 9. (10 points) Use change of variablesto evaluate the integral , where
• 7. Use the transformation z = u/v, y = v to evaluate the doubleintegralwhere R is the region in the
• 9.銀行年利率為 1%,若存進10萬元,以月計算複利,大約_______年後存款會變成40萬元(在個位數四捨五入,差距在2年内都算對)
• 1. Let f:R→R be twice differentiable. If f" is nowherevanishing, then f has at most two distinc
• (F). The coefficient of x9 in the Taylor expansion of fo dt is________.
• 29. 橢圓Γ： 上任一點P，到直線L：x−2y＋15＝0的距離之最小值為(A) (B) (C) (D)
• 10.(本大题請標出底下各小前的編號)f(x)=,回答下列問題,並作圖:【題組】(1)此函數圖形遞減凹向下(convace down)的範圍在________
• 2. Prove that
• 1. 小明欲配製0.5M的碳酸鈉溶液1升，則其配製的方法，下列何者正確？(碳酸鈉分子量為106)(A)將53克的碳酸鈉溶入1升的水中(B)將106克碳酸鈉溶於水中，並定量至1升(C)將106克的碳酸鈉