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Unformatted text preview: MATH 140Final Examination Dec. 15, 2005
INSTRUCTIONS: Write your name, section number and TA’s name on each answer sheet. Answer
each numbered question on a separate answer sheet. You must show all appropriate work in order to receive credit for an answer. At the end of the exam, arrange the answer sheets in order and write and
sign the honor pledge on the first one (only on the ﬁrst one). No Calculators or other electronic devices may be used during the exam. 1. Compute each of the following limits as a real number or 00 or —00 or state ” does not exist”. sin(x + 1) (b) lim 3:2 + 4x (a) hm x2 __ 1 wa4 m x—>—1 2. Find each of the following. It is not necessary to simplify your answers.
(a) Let f(t) = x/t sin(5t). Find f’(t).
(b) Let y = . 3 + 629”
3. (a) Evaluate as a number, 00, or —002 11m ——
:c—a—oo 1 —— 56’295 (b) Let 9(6) = ln(sec 30). Find g’(0) 4. (a) Find an equation for the tangent line to the graph of the equation $23; + 3342 = 2 at the point
(1,1).
(b) The velocity U of an object at time t is 11(t) = 6 sin(3t) — 4cos(3t). Find the position p(t) of the
object at time t if p(0) = 3. 5. A 15—foot ladder leans against a wall. The bottom of the ladder is pulled away from the wall so that
the top of the ladder remains on the wall and when the bottom is on feet away from wall, it is moving
at a rate of x / 2 feet per second. How fast will the top of the ladder be moving down the wall when
the top is 5 feet above the ground? (Note: 152 == 225) 6. Evaluate each of the following. If you use a substitution in an integral, state it clearly. If the
substitution is used in a deﬁnite integral, change the limits of the integral when you change the
variable. (a) / (at—53+???) dt 7. Suppose f and g are continuous on the interval [1, 6] with f(1) = 2, f(6) = 4, f’(1) = 1.5, f’(6) = ——2,
ff f(ac)d:t = 13, ff f(:c)d:c = 5 and f169(x)da: = —5. Find each of the following: 4 6 x _
(a)/1 ﬁx)“: (bl/1 (10f($)—g(x))dx ___f( ) 2 x~1 3/2
(b) / 4xx/2m + 1 dm
0 (0) $1311 6
@Afmm Please turn over for the remaining problems. (10 each) (Spts) (IOpts) (10pts) (8M3) (10pts) (10pts) (15pts) (10 each) (16pts) C 8. Minnie Max used tables of signs to ﬁnd the following number lines for the ﬁrst and second derivatives (25pts)
of a function f that is continuous on (—oo,oo). Using these number lines answer the following questions. (00 in the 1" number line indicates a vertical tangent line; U indicates that f” is undeﬁned
at 0.) Justify your answers. f’—~0++oo++ f”++U~—0++ _——_l——————— —————————l————
—1 0 0 2 (a) On what interva1(s) is f increasing? decreasing?
(b On what interval(s) is the graph of f concave upward? concave downward?
(c At what value(s) of a: does f have a relative maximum value? a relative minimum value? )
) (d) Where does the graph of f have inﬂection point(s)?
) (e If f(4) = 0 = f(0), f(x) = 00, and Inn f(a:) = 00, draw a graph for f. {II+00 9. (a) Let = 3:3 + m2 + 2. Find an integer It so that f has a zero in the interval [16, k + l]. Justify (9pts)
your answer (including the theorem used). (b) Let f be the function in part (a). Use a tangent line approximation to estimate f (2.01). (9pm) 10. (a) Let = x3 — 4:0 and 9(93) = m2 + 29:. Draw the graphs of f and g on the same coordinate
axes. Label the graphs f and 9. Label the mintercepts and points of intersection with their
coordinates. Find the alt—coordinates of the relative extreme points. (Hint: There are three
points of intersection.) (7pts) (b) Set up the integral(s) needed to ﬁnd the area A of the bounded region R between the graphs of
= m3 — 4x and y = 3:2 + 23:. Do not use absolute values and do not integrate. (8pts) 11. You must construct a closed rectangular box with a volume of 72 cubic feet and its bottom twice as (15pm)
long as it is wide. Find the dimensions of the box that minimizes its total surface area. End of Examination “Tr ...
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 Spring '08
 Gulick
 Derivative, separate answer sheet

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