Download - Math 1A, Exam #3 - Berkeley Engineering

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Page 1: Math 1A, Exam #3 - Berkeley Engineering

Math 1A, Exam #3 1. Find the following limits:

(a) lim!β†’βˆ©

(𝑒! + π‘₯)!/! (b)

lim!β†’!!!

tan! π‘₯ βˆ’ 37 tan! π‘₯ + sin π‘₯

(c)

  lim!β†’!!

3π‘₯ + 2π‘₯! + 7

2. Find the following derivatives:

(a)

 π‘‘𝑑π‘₯

cosh!! π‘₯

(b)

   π‘‘𝑑π‘₯  ln(cosh π‘₯!)𝑒! + 2

3. Find the slope of the curve 𝑦! = π‘₯! + 8𝑦 βˆ’ 9,  at the point (1,2)

(a) Find 𝑦” at (1,2) 4. Use differentials to estimate tan !

!+ 0.05 .

5. State fully and carefully the Mean Value Theorem. (a.) Let 0 < π‘Ž < 𝑏. Show that there exists a number c such that                          π‘Ž < 𝑐 < 𝑏 and 𝑐 ln !

!= 𝑏 βˆ’ π‘Ž.

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6. A cylindrical gob of goo is undergoing a transformation in which its height is decreasing at a rate of 1cm/sec, while its volume is decreasing at the rate of 2πœ‹    π‘π‘š!/sec. (It retains its cylindrical shape while all this is happening.) If, at a given instant, its volume is 24πœ‹  π‘π‘š! and its height is 6cm, determine whether its radius is increasing or decreasing at that instant, and at what rate. 7. Determine the maximum and minimum values of the function 𝑓 π‘₯ = π‘₯ βˆ’ 3π‘₯! ! on the interval [-1, 27], find all points (x-values) where they occur. 8. Find all asymptotes (horizontal, vertical, and slant) of the function,

𝑓 π‘₯ =π‘₯! βˆ’ 1π‘₯! βˆ’ 1

9. A cyclist traveling at 30 ft/sec decelerates at a constant 3ft/𝑠𝑒𝑐!. How many feet does he travel before coming to a complete stop? 10. If Newton’s method is used to solve π‘₯! + π‘₯! + 2 = 0 with an initial approximation π‘₯! = βˆ’2, what is the second approximation, π‘₯!? 11. Find the following integrals: (a)

tanπœ‹4π‘₯!

!

!!𝑑π‘₯

(b)

𝑑π‘₯9π‘₯! + 1

(c)

π‘₯! + 1π‘₯!

𝑑π‘₯

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(d)

π‘₯ sin π‘₯! 𝑒!"# !!𝑑π‘₯

(e)

π‘₯   π‘₯ βˆ’ 1!!

!𝑑π‘₯

12. Find the integral π‘₯!𝑑π‘₯!

! using the definition as a limit of

Riemann sums. You may use any (valid) choice of π‘₯!βˆ—, and may also

use any of the formulas,

1!

!!!

= 𝑛

𝑖 =𝑛(𝑛 + 1)

2

!

!!!

𝑖!!

!!!

=𝑛(𝑛 + 1)(2𝑛 + 1)

6

𝑖! =𝑛(𝑛 + 1)

2

!!

!!!

13.The region bounded by the curves 𝑦 = 2π‘₯! and 𝑦 = 3π‘₯ βˆ’ π‘₯! is rotated about the y-axis. Find the volume of the resulting solid.

(a) The same region is rotated about the line 𝑦 = βˆ’1. Write an integral expressing the volume of the resulting solid. (DO NOT evaluate it.)