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CALCULUS I - FINAL EXAM
December 12, 2001
Show all work for full credit. You may use a calculator, but do not use books or notes. The point
value of each problem is given in the left-hand margin. You have two hours.
(18) 1. Evaluate the following limits algebraically.
x3 − 4x
a) lim
=
x→2 x − 2
tan h
=
h→0 2h
b) lim
x2 − x3
=
x→∞ 3x3 + 4x2
c) lim
(8) 2. State the definition of the derivative of a function f (x) as a limit.
page 2 of 10
(20) 3. Compute the following derivatives. DO NOT SIMPLIFY
a) f 0 (t) where f (t) =
b)
q
(2t + 1)3 − 5 .
d tan3 (x)
=
dx x + 1
(10) 4. Find the points where the tangent line to the curve x2 + y 4 = 5 at (2, 1) intersects the x
and y axes.
page 3 of 10
(10) 5. Let f (x) = x(2 − x)1/3 . Answer the following questions given that
f 0 (x) =
6 − 4x
,
3(2 − x)2/3
f 00 (x) =
4x − 12
.
9(2 − x)5/3
a) For what values of x is f (x) defined?
b) Determine the open interval(s) on which f (x) is increasing.
c) Determine the open interval(s) on which the graph of f (x) is concave up.
(12) 6. The graph of y = f (x) is given below.
6
•
-2
•
1
@
@
@
@
◦
@ 1
@•
-1
@ @
2
@
R
@
a) Find lim+ f (x) =
b) Indicate all values of x at which f (x) is not continuous.
c) Indicate all values of x at which f 0 (x) is not defined.
d) Evaluate f 0 (−2) =
e) Evaluate
R1
−2
f (x) dx =
x→1
page 4 of 10
(10) 7. A cylindrical can of volume 6π cubic inches is constructed out of two materials. The
material for the side wall costs 10 cents per square inch while the two lids cost 15 cents per
square inch. How should the can be constructed so as to minimize the total cost? Determine
the radius r and height h . (Recall, V = πr2 h and A = 2πrh for the side wall).
(10) 8. Evaluate the definite integral
Z
0
2
(y + 2) dy
√ 2
y + 4y + 1
page 5 of 10
(10) 9. Find the position function s = s(t) of an object moving along a straight line given that
its velocity v = s0 (t) = 4t3 − 3t2 (m/sec) with t measured in seconds and s(2) = 1 meter.
(10) 10. Car A is travelling east (to the right) along the x -axis while car B is travelling south
(down) along the y -axis. At a particular instant, car A is at the point (40, 0) (mi) and
travelling east at 50 mi/hr while car B is at (0,30) and travelling south at 30 mi/hr. At what
rate is the distance between the two cars changing at this instant?
page 6 of 10
(14) 11. Sketch the graph of the function g(x) = x3 (x − 1)2 as follows.
a) Give the exact values of all critical points and draw a number line showing where g(x) is
increasing and decreasing.
b) Given that g 00 (x) = 2x(10x2 − 12x + 3) determine all inflection points. Just give the exact
x -coordinates.
c) In the table below, give both coordinates of all local minima, maxima and inflection points.
Round your answers to two decimal places with the help of a calculator. Then sketch the graph
of g(x) .
6
x
y = g(x)
.15
.10
.05
-1
1
−.05
−.10
−.15
?
page 7 of 10
(16) 12. Evaluate the indefinite integrals.
Z √
a)
sin x cos x dx =
b)
Z
(cos(3t) + sec(2t) tan(2t)) dt =
(8) 13. Find the linear approximation L(x) to the function f (x) =
√
x near x = 9 .
page 8 of 10
R7
Pn
(8) 14. Estimate 3 sinx x dx using a Riemann sum S = i=1 f (x∗i )∆x with n = 4 , a regular
partition (all intervals of the same length), and x∗i chosen as the left endpoint of the ith
interval. (Leave your answer as a sum of values involving the sin function.)
(10) 15. Make a sketch of the region bounded between the parabolas y = x2 and y = 8 − x2 and
then calculate its area.
page 9 of 10
(12) 16. Below is a sketch of the region bounded by the curve y = x − x3 and the x-axis over
the interval 0 ≤ x ≤ 1 . Set up integrals for the following volumes but do not evaluate the
integrals.
y 6
•
1
x
a) The volume of the solid obtained by rotating the region around the horizontal line y = −1 .
b) The volume of the solid obtained by rotating the region around the y -axis.
page 10 of 10
(7) 17. Set up the integral that gives the length of the arc y = x4/3 , 0 ≤ x ≤ 8 . Do not evaluate
the integral.
(7) 18. Set up the integral that gives the surface area of the surface generated by rotating the arc
in problem 17 about the x -axis. Do not evaluate the integral.
(5) 19. Extra Credit. Second Order Approximations. On the road to Calc II: A second order
approximation to a function f (x) near x = x0 is a quadratic function p(x) = ax2 + bx + c
having the property that the two functions f (x) and p(x) have the same value, the same
slope and the same concavity at x = x0 . Its graph is a parabola that best approximates the
curve y = f (x) near x = x0 . Find the second order approximation to f (x) = cos x near
x = 0.