6. (10 points) Evaluate the infinite sum

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

(b)dx.

3. (10 points) Evaluate |(0,1) foru(x,y) = (eˣ - dt.Note: Answer must in numerical expression and natural constants like π, e, ...etc.

4. (10 points) Evaluate |(2,0) foru(x,y) = h(x² + y², 3x – 4y)whereh(s,t) =

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

7. (10 points) A dog is running along a semi-circular track with radius 1 km incounterclockwise 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) forf(x) = x⁵ + 2x³ + 2x.

9. (10 points) Evaluatedydx.

10. (10 points) Use Lagrange multiplier to find the extreme value off(x, y, z) = exyzsubject to the constraintx³ - y² + z³ = 24.Also, indicate the value(s) you obtain is (are) maximum or minimum.

IV. WritingNowadays, communicating online seems to become the norm. People are writing blogs, creating Facebook profiles, sharing videos on YouTube and tweeting on Twitter. In particular, social media has given people the power to share and stay connected； however, social media may also promote negative experiences such as feeling inadequate about your life or appearance. Please write an essay of about 300 words to illustrate how people can minimize the negative effects of social media.

(a) (5 points)