112年 - 112 國立臺灣大學_碩士班招生考試:微積分(C)#114173

試卷資訊

選擇題 (0)

申論題 (10)

(a) (10%) Given,express x= az₁+β and γ=yz₁ +δz₂+εte such that What are a, β, γ, δ, and ε?

(b) (15%) Evaluate the expectation E[exp(x + y)I{x ≤ 0}], where I{A} is an indicator function returning 1 (0) ifthe condition A is true (false), and express the calculation result a +bn₁(c;0,1)+dN₁(e;0,1)+fn₁(g:0,1)n₁(h;0,1)+ in1(j；0,1)N₁(k;0,1)+ IN₂.What are a, b, c, d, e, f, g, h, i, j, k, I, m, and n?

(a) (10%) Solve both of these differential equations and express D(t) and I(t) in terms of a, r, l₀, and D₀.

(b) (5%) Determine what happens to this ratio in the long run by computing the limit

3. (10%) IB company produces x units of commodity A and y units of commodity B. All the units can be sold for p = 20 - 5x dollars per unit of A and q = 4 - 2y dollars per unit of B. The cost (in dollars) of producing these units is given by the joint-cost function C(x,y) = 2xy + 4. What should x and y be to maximize profit?

4. (10%) Find for the equation y³ +y² -5y -x² = -4.

5. (10%) Evaluate dx.

6. (10%) Find the value of (csc4x-cot4x)dx .

7. (10%) Use a Maclaurin polynomial to find the value of accurate to four decimal places.

8. (10%) Given f(x,y)= e^x siny+ In xy, find .