198 STEP 4. Review the Knowledge You Need to Score High- The position function of a moving particle
on a line iss(t)=sin(t) for 0≤t≤ 2 π.
Describe the motion of the particle. - A coin is dropped from the top of a tower
and hits the ground 10.2 seconds later. The
position function is given as
s(t)=− 16 t^2 −v 0 t+s 0 , wheresis measured
in feet,tin seconds, andv 0 is the initial
velocity ands 0 is the initial position. Find
the approximate height of the building to
the nearest foot. - Find the equation of the tangent line to the
curve defined byx=cost−1,
y=sint+tat the point wherex=
− 1
2
.
- An object moves on a path defined by
x=e^2 t+tandy= 1 +et. Find the speed of
the object and its acceleration vector with
t=2. - Find the slope of the tangent line to the
curver=3 sin 4θatθ=
5 π
6
.
- The position of an object is given by〈
30 t, 25 sin
t
3
〉. Find the velocity and
acceleration vectors, and determine when
the magnitude of the acceleration is equal
to 2.
25. Find the tangent vector to the path defined
byr=
〈
lnt,ln(t+4)〉
at the point where
t=4.9.7 Cumulative Review Problems
(Calculator) indicates that calculators are
permitted.
- Find
dy
dx
ify=xsin−^1 (2x). - Givenf(x)=x^3 − 3 x^2 + 3 x−1 and the
point (1, 2) is on the graph off−^1 (x). Find
the slope of the tangent line to the graph
off−^1 (x) at (1, 2). - Evaluate limx→ 100
x− 100
√
x− 10
.
- A functionfis continuous on the interval
(−1, 8) withf(0)=0,f(2)=3, and
f(8)= 1 /2 and has the following
properties:
INTERVALS (−1, 2) x=2 (2, 5) x=5 (5, 8)
f′ + 0 −−−
f′′ −−− 0 +(a) Find the intervals on whichfis
increasing or decreasing.
(b) Find wheref has its absolute extrema.
(c) Find wheref has the points of
inflection.
(d) Find the intervals on whichfis
concave upward or downward.
(e) Sketch a possible graph off.- The graph of the velocity function of a
moving particle for 0≤t≤8 is shown in
Figure 9.7-1. Using the graph:
(a) Estimate the acceleration when
v(t)=3 ft/s.
(b) Find the time when the acceleration is
a minimum.