This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: MIDTERM 1
MATH 19 A 5/5/2006 Instructor: Frank Béiuerle, Ph.D. Your Name: _______________________________ Your TA: _—_——_——————.————_—_——.——_——_—.—_—._———.—— Max Your score Problem 1:“ 15 Problem 2: ' 10
Problem 3: 10
Problem 4: 20
Problem 5: 25
Problem 6: 10
Problem 7: 10
TOTAL: 100 Good Luck! 1. (15 points) (a) Give an informal description of what it means that ” f (at) is continuous at J: = a”. (b) Give a formal deﬁnition (using linis) of ” f (:3) is differentiable at :c = a”. (c) Use the limit deﬁnition of the derivative to compute the derivative of f (:5) = $2 + 1 at
5:: = 2. (d) Use the appropriate diiferentiation rules to compute the same derivative again to verify
your answer in (c). i’. (10 points) Assume 5(t) = t3  121E + 5 is the position of an object in a onedimensional
system. (a) What is the velocity of the object at time t = 0? (b) What is the speed of the object at time t = 0? Explain the difference to your answer in (a). (c) At what time(s) is the object at rest? 3. (10 points) Use the given graph of f (cc) to carefully sketch the graph of f’(a:) in a new
coordinate system.
’19 ‘6 Hint: Find/estimate f’(1),f’(l),f’(0) and think about what it means for f[:c) when
f'($) > U or when f’(:L') < O. sin 3: 4. (20 points) Compute the following limits. Justify steps. You may use that iim0 = 0
3—4» a:
sin2 a:
1.
(a) :cl—I—Iilo 2;;
l _ 3
(b) lim _+_$i =—*°° 1 +z2 , tanh
(c) blinoT
(d) 1i1n s_1n_t t—>0+ \ﬂ 5. (25 points) Compute the requested derivatives of the functions: (NO NEED TO SIM
PLIFY) dy
(8—) Ex’ for y — x/5 (b) y' for y =x2+cosm+secx—3 (c) f’(x) for M) = —f§ df x
(d) dx for f(:1:)—2:2 tan:
d?! 5 4 9 2 a
(e) Efory=(z ——:L' +3—2)(5:r: ’13:: —:1: +3—12) 1' 6. (10 points) Compute an equation of the tangent line to the curve 3; = at the point 8
Pl.
(’2) 3+1 7. (10 points) Find all real number values for a so that the function f(:c) = ax? —— (12.17 + 1 has
a horizontal tangent line at :c = 1. ...
View
Full Document
 Spring '06
 Bauerle

Click to edit the document details