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(MATH1013)[2012](f)midterm~=791l^_77782.pdf
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Math1013 Calculus I
Midterm Exam Solution (White-Green Version)
Part I: Multiple Choice Questions
MC Answers: White-Green Version
Question 1 2 3 4 5 6 7 8 910 Total Answer cdc eabbde e
MC Answers: Orange-Yellow Version
Question 1 2 3 4 5 6 7 8 910 Total Answer d cd cd eb a cb
Solution (White-Green Version.)
1. Which of the following intervals is the domain of the function de.ned by the formula f(x)= x2 .1
?
3
xlnx +1
(a) (0,) (b) (e,) (c) (1
,) (d) (.1
,) (e) (0,e)
ee
Solution
The answeris(c), sincelnx +1 > 0 .. x>e.1 .
2. Across its whole domain, which of the functions given by the following graphs has an inverse function?
y
y
(d)
(e)
x
x
Solution
The answeris(d): thegraphpassesthehorizontallinetest, orthefunctionis one-to-one onits domain.
.4x2 +2x
3. Find the limit lim
x1 4x2 .8x +3
2
1
(a) .1(b) 0 (c) (d) 1 (e) Does not exist
22
Solution
.4x2 +2x .2x(2x .1) .2x 1
Theansweris(c):lim =lim =lim =
x1 4x2 .8x +3 x1 (2x .1)(x .3) x1 2x .3
2
22 2
250t +80 t
4. The temperature function of an object at t 0 hours is given by T(t)= ,in .C. As
25+5t
t , approximately to what steady temperature will the object approach?
(a) 0.C (b) 10.C (c) 16.C (d) 40.C (e) 50.C
Solution
250+80 1
250t +80 t t
Theansweris(e):lim =lim 25 =50
t 25+5t t +5
t
5. Using the graph of f given below, .nd lim f(g(x))where g(x)=1.2x.
x1+
approach -1 from smaller than -1 from 1+
(a) -2 (b) 0 (c) 1 (d) 3 (e) does not exist
Solution
The answeris(a): As x 1+ , g(x).1., hence f(g(x)).2.
x2 + x
6. Find lim + .x .
x. x
(a) . (b) (c) -1 (d)1 (e)0
Solution
.
x2 + x 1
Theansweris(b): lim + .x = lim . 1+ + .x = .
x. x x. x
7. Find the equation of the tangent line to the graph of the function y = f(x)= .6x2 +3x at the point(2,f(2)).
(a) y = .15x +12 (b) y = .21x +24 (c) y =21x .60
(d) y = .12x .6 (e) y =15x .48
Solution
The answeris(b): f (x)= .12x +3, f (2)= .21, f(2)= .18.
Tangent line equation y = .21(x .2).18 = .21x +24.
sin x
8. Compute g (1)for g(x)= e .
(a) e (b) (c) .e (d) . (e) e
Solution
sin x sin
The answeris(d): g (x)= e cosx, g (1)= ecos = ..
9. If f(x)= g(x2 +7) and g(x)= x2/3, compute the derivative f (1).
(a) .2 (b) .1 (c) 1 (d) 1 (e) 2
3 22 33
Solution
22
The answeris(e): f (x)= g (x2 +7) 2x, g (x)= x.1/3, hence f (1)= g (8) 2= 2= 3 .
3381/3
10. The number of gallons of water in a tank t minutes after the tank started to drain is V(t)= 25(30.2t)2 . How fast is the water running out at the end of 8 minutes?
(a) 250 gal/min (b) 500 gal/min (c) 700 gal/min (d) 1000 gal/min (e) 1400 gal/min
Solution
The answeris(e): V (t)= .100(30.2t),V (8)= .1400 gal/min.
Part II: Answer each of the following questions.
(x)
11. [10 pts] The graphs ofy =