Review Exam 3 MAC 2311 ( Calculus I)
3.4, 3.5, 5.3, 4.1, 4.2, 4.3, 4.4, 4.8
03/24/2015
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1) Verify that the hypotheses of Rolle’s Theorem are satisfied and find all values of c in that interval
that satisfy the conclusion. a) f (x) = x3 − 3x2 + 2x, [0, 2]
b) g(x) = cos x, [π/2, 3π/2]
c)
h(x) = ln(4 + 2x − x2 ), [−1, 3]
2) Verify that the hypotheses of the Mean-Value Theorem are satisfied and
√ find all values of c in that interval
c) h(x) = x − x1 ,
that satisfy the conclusion. a) f (x) = x3 + x − 4, [−1, 2]
b) g(x) = x + 1, [0, 3]
√
[3, 4]
d) i(x) = 25 − x2 [−5, 3]
3) Tell whether the function f (x) = tan x satisfies the hypotheses of Rolle’s on [0, π]. Justify your answer
4) Tell whether the function f (x) = x2/3 satisfies the hypotheses of the MVT on [−1, 8]. Justify your answer
10) Use the MVT to show that | sin x − sin y| ≤ |x − y| for all real values of x and y.
√
5) Find the local linear approximation of a) arctan x at π/3, b) 4x + 1 at 2
√
6) Use local linear approximation (round off your answer to 5 decimal places) a) 8, b) cos(63◦ ) c) ln(2.705)
√
7) Find dy, ∆y when a) y = x 2x + 5, b) y = x cos x
8) a. Find the domain and all asymptotes (if any) b. determine the intervals of increase and decrease
c. Find the intervals on which the function is concave upward
d. Sketch the graph
e.√ Find the
coordinates of relative min or max or inflection points. p) p(x) = 2 − x + 2x2 − x3 q) q(x) = 3 x2 − 3
g) g(x) = x2/3 ex
r) r(x) = 4x1/3 − x4/3
h) h(x) = x2 ln x, f) f (x) =
9) Find all relative extrema over (0, 2π) for a) y =
√
ln(2x + 1)
2x + 1
3 + sin(2x) b) f (x) =
sin x
2 − cos x
10) The graph of f 0 is given. a) What are the intervals of increase of f ? b) On what intervals is f concave
down? Sketch a possible graph for f. repeat a), b) with the graph of f 00
11) Find the indicated integral
Z
Z
Z y
Z
√
1
e − e−y
x2
√
a)
dx
b)
dy
c)
2x
3
+
2x
dx
d)
dx
2
y
−y
4x + 9
e +e
1 − x3
Z
Z
Z
Z
t
x4 + 3x3 − 2x − 7
3
4
e)
dt
f)
)
dx
h)
cos
(3θ)
sin
(3θ)
dθ
g)
(
tan4 (2θ) sec4 (2θ) dθ
t4 + 1
x2 + 1
√
Z
i)
t
√
dt
3t + 4
Z
j)
4
cos θ dθ
Z
k)
4
Z
cos θ dθ
l)
e
√
2x+1
2x + 1
dx
12) Solve the initial value problem for y = f (x)
a)
dy
= x(3x − 2)5 where y = 2 when x = 1
dx
c) f 0 (x) = (1 + sin x)9 cos x where f (π/6) = 1
b)
dy
1
√ where y = −3 when x = 4
=
dx
(x + 1) x
section 3.4
13) A particle is moving along the curve y =
x
. Find all values of x at which the rate of change of x
+1
w.r.t time is three times that of y(assuming dx/dt is never 0)
x2
14) A particle is moving along the curve 16x2 + 9y 2 = 144. Find all points (x, y) at which the rates of change
of x and y with respect to time are equal.(Assume that dx/dt and dy/dt are never both zero at the same
point)