Math 120 Sample questions for Test 2, with answers

Transcription

Math 120 Sample questions for Test 2, with answers
```Math 120
Sample questions for Test 2, with answers
Not all questions will necessarily be like the ones on the following pages,
but they do give a pretty good idea of the types of questions to expect.
Answers (but not complete solutions) follow the set of questions. Do not
consider #s 6 & 7.
Also, study Quizzes 4,5,6 which covered §§ 3.1–7 for which you were
provided detailed solutions. There will be at least one Test 2 question
similar to a question on each of these quizzes. In general, though, expect
some test questions to be a bit more challenging than quiz questions.
Finally, on the last page is a practice quiz that includes a question from
each of §§ 3.8 & 3.9, with solutions included. (Same comment as above
about the relative difficulty of quiz and test questions.) (The first question,
on §3.7, appeared on our recent Quiz 6.)
Stewart - Calculus ET 6e Chapter 3 Form A
1. If f (3) = 4, g (3) = 2, f ′(3) = −5, g ′(3) = 6 , find the following numbers.
( f + g )′(3) = __________
( fg )′(3) = __________
( f / g )′(3) = __________
′
§ f ·
¨¨
¸¸ (3) = __________
2.
Find the points on the curve y = 2 x 3 + 3 x 2 − 12 x + 1 where the tangent is horizontal.
3.
Find the equation of the tangent to the curve at the given point.
y = 1 + 4sin x , (0,1)
4.
Differentiate.
g ( x) = x 7 cos x
5. Find f ′ in terms of g ′ .
f ( x) = x 2 g ( x)
6.
If a ball is thrown vertically upward with a velocity of 200 ft/s, then its height after t seconds is
s = 200t − 10t 2 .
What is the maximum height reached by the ball?
7.
Find the limit.
lim
x →π / 4
8.
sin x − cos x
cos 2 x
Calculate y ′ .
y = x cos x
9.
A spherical balloon is being inflated. Find the rate of increase of the surface area S = 4πr 2 with
respect to the radius r when r = 1 ft.
10. Differentiate the function.
f ( x) =
7
x5
Stewart - Calculus ET 6e Chapter 3 Form A
11. Find an equation of the tangent line to the curve.
y=
x
at (4, 0.2)
x+6
12. A plane flying horizontally at an altitude of 4 mi and a speed of 465 mi/h passes directly over a radar
station. Find the rate at which the distance from the plane to the station is increasing when it is 10 mi
away from the station. Round the result to the nearest integer.
13. Find the limit if g ( x ) = x5 .
lim
x→ 2
g ( x) − g (2)
x−2
14. A company makes computer chips from square wafers of silicon. It wants to keep the side length of a
wafer very close to 16 mm. The area is A(x). Find A′(16).
15. Calculate y ′ .
xy 4 + x 2 y = x + 3 y
16. Find the first and the second derivatives of the function.
y=
x
3− x
17. If f (t ) = 4t + 1 , find f ′′(2) .
18. If y = 2 x 3 + 5 x and
dy
dx
= 3, find
when x = 5.
dt
dt
19. The volume of a cube is increasing at a rate of 10 cm3 / min . How fast is the surface area increasing
when the length of an edge is 30 cm .
20. If f (t ) =
18
3+t2
find f ′(t ).
Stewart - Calculus ET 6e Chapter 3 Form A
1.
1, 14, -8.5, 8.5
2.
(1, -6), (-2, 21)
3.
y = 2x + 1
4.
dg ( x)
= 7 x 6 cos( x) − x 7 sin( x)
dx
5.
f ′( x) = 2 xg ( x) + x 2 g ′( x)
6.
1000
7.
−
8.
1 § x sin x − cos x ·
y′ = − ¨
¸¸
x
¹
9.
8π
10.
df
5 7
=− 6
dx
x
11.
y=
12.
456
13.
80
14.
32
15.
y′ =
16.
3(3 − x )−2 , 6(3 − x )−3
17.
f ′′(2) = −
18.
19.
20.
2
2
1
(x − 4) + 0.2
200
1 − y 4 − 2 xy
4 xy 3 + x 2 − 3
4
27
465
4 2
cm / min
3
−36t
(3 + t )
2 2
```