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MATH1013 Calculus I, 2013-14 Fall
Week 08 Seventh tutorial Worksheet: Applications
Name:
ID No.:
(L10, T11A)
Tutorial Section:
Complete at least TWO questions from the following questions.
(Solution of this worksheet will be available at the course website the week after.)
1. Demonstration (page 111, Q 84) Use the Intermediate Value Theorem to verify that the following
equations have three solutions on the given interval. x3 + 10x2 − 100x + 50 = 0; (−20, 10)
2. Demonstration (page 111, Q 88) Suppose you park your car at a trailhead in a national park and
begin a 2-hr bike to a lake at 7 a.m. on a Friday morning. On Sunday morning, you leave the lake at 7
a.m. and start the 2-hr hike back to your car. Assume the lake is 3 mi from your car. Let f (t) be your
distance from the car t hours after 7 a.m. on Friday morning and let g(t) be your distance from the car
t hours after 7 a.m. on Sunday morning. a. Evaluate f (0), f (2), g(0), g(2). b. Let h(t) = f (t) − g(t).
Find h(0) and h(2).
3. (Demonstration) (page 178, Q. 27) A stone is thrown vertically into the air at an initial velocity of
96ft/s. On Mars, the height s (in feet) of the stone above the ground after t seconds is s = 96t − 6t2 ,
and on Earth, s = 96t − 16t2 . How much higher will the stone travel on Mars than on Earth?
4. (Demonstration) (p.216 Q. 10, 12) Differentiate (a) arcsin(ln x) and (b) arcsin(esin x ).
5. (Demonstration) (p.217, Q. 47) Find the derivative (f −1 )0 (3) if f (x) = x3 + x + 1.
6. (Demonstration) (p.198, Q. 74 ) The
√ lateral surface area of a cone of radius r and height h (the surface
area excluding the base) is A = πr r2 + h2 . a. Find dr/dh for a cone with a lateral surface area of
A = 1500π. b. Evaluate this derivative when r = 30 and h = 40.
7. (Class work) (page 178, Q. 26) On Moon, a feather will fall to the ground at the same rate as a heavy
stone. Suppose a feather is dropped from a height of 40 meter above the surface of the moon. Then
its height s (in meters) above ground after t seconds is s = 40 − 0.8t2 . Determine the velocity and
acceleration of the feather the moment it hits the surface of the moon.
Answer
8. (Class work) (page 178, Q. 29) A stone is thrown from the edge of a bridge that is 48ft above the
ground with an initial velocity of 32ft/s. The height of the stone t seconds after it is thrown is f (t) =
−16t2 + 32t + 48. If a second stone is thrown from the ground, then its height above the around after
t seconds is given by g(t) = −16t2 + v0 t, where v0 is the initial velocity of the second stone. Determine
the value of v0 so that both stones reach the same high point.
1
Answer
9. Classwork (page 111, Q 85) Use the Intermediate Value Theorem to verify that the following equations
have three solutions on the given interval. 70x3 − 87x2 + 32x − 3 = 0; (0, 1)
Answer
10. Classwork (page 112, Q 100) Let f (x) = |x|
x . Then f (−2) = −1 and f (2) = 1. Therefore, f (−2) <
0 < f (2), but there is no value of c between -2 and 2 for which f (c) = 0. Does this fact violate the
Intermediate Value Theorem? Explain.
Answer
11. Classwork (page 198 Q 76) The volume of a torus (doughnut or bagel) with an inner radius of a and
an outer radius of b is V = π 2 (b + a)(b − a)2 /4. a. Find db/da for a torus with a volume of 64π 2 . b.
Evaluate this derivative when a = 6 and b = 10.
Answer
2