109年 - 109 台灣綜合大學系統_學士班轉學考聯合招生考試：微積分B#124117

(a)

(b)

(a) x sec²x dx.

(b)dx.

3. (10 points) Evaluate |_(0,1) for
u(x,y) = (eˣ - dt.

Note: Answer must in numerical expression and natural constants like π, e, ...etc.

4. (10 points) Evaluate |_(2,0) for
u(x,y) = h(x² + y², 3x – 4y)
where
h(s,t) =

5. (10 points) Find the volume of the solid generated by rotating the curve
y = sin(x²)
over 0 ≤ x ≤ 1 around the y-axis.

6. (10 points) Evaluate the infinite sum

by some manipulations of the Taylor series of f(x) = xeˣ.

7. (10 points) A dog is running along a semi-circular track with radius 1 km in
counterclockwise direction with speed 0.1 km per minute (See Figure).

Let h be the distance between the dog and point A. Find the rate of change of h at point D, half way between B and C.

8. (10 points) Find (f⁻¹)'(5) for
f(x) = x⁵ + 2x³ + 2x.

9. (10 points) Evaluate
dydx.

10. (10 points) Use Lagrange multiplier to find the extreme value of
f(x, y, z) = e^xyz
subject to the constraint
x³ - y² + z³ = 24.
Also, indicate the value(s) you obtain is (are) maximum or minimum.