Question

1. a True or False? If ∫ [ f ( x ) ⋅ g ( x ) ] d x = [ ∫ f ( x ) d x ] ⋅ [ ∫ g ( x ) d x ]. Justify your answer.

B. Find ∫ 0 π 4 sec 2 θ tan 2 θ + 1 d θ

C. Show that ∫ 0 π 2 sin 2 x d x = ∫ 0 π 2 cos 2 x d x.

D. Find the indefinite integral: ∫ ( sin 3 x ) ( c o s x ) d x

E. **Using substitution**: Describe why
∫ x ( 5 − x 2 ) 3 d x ≠ ∫ u 3 d u where u = 5 − x 2.

Answer #1

Find the values of sin x, tan x, cot x, sec x, and csc x and cos
x given that cos(2x)=(3/5) and 3 π /2 < 2x <2 π

1.) Let f ( x , y , z ) = x ^3 + y + z + sin ( x + z ) + e^( x
− y). Determine the line integral of f ( x , y , z ) with respect
to arc length over the line segment from (1, 0, 1) to (2, -1,
0)
2.) Letf ( x , y , z ) = x ^3 * y ^2 + y ^3 * z^...

For functions f and g, where f(x) = 1 + x 2 , g(x) = x − 1 in
C[0, 1] with the inner product defined by the integral: hf , gi = Z
1 0 f(x)g(x)dx, (a) find the norm of f and g. (b) find unit vectors
in the directions of f and g. (c) find the cosine of the angle θ
between f and g. (d) find the orthogonal projection of f along
g

Define g(x) = f(x) + tan-1 (2x) on [−1, √3/2 ].
Suppose that both f'' and g'' are continuous for all x-values on
[−1, √3/2 ]. Suppose that the only local extrema that f has on the
interval [−1, √3/2 ] is a local minimum at x = 1/2 .
(a) Determine the open intervals of increasing and decreasing
for g on the interval [1/2 , √3/2] .
(b) Suppose f(1/2) = 0 and f(√3/2) = 2. Find the absolute...

1a.
Find the domain and range of the function. (Enter your answer
using interval notation.)
f(x) = −|x + 8|
domain=
range=
1b.
Consider the following function. Find the composite
functions
f ∘ g
and
g ∘ f.
Find the domain of each composite function. (Enter your domains
using interval notation.)
f(x) =
x − 3
g(x) = x2
(f ∘ g)(x)=
domain =
(g ∘ f)(x) =
domain
are the two functions equal?
y
n
1c.
Convert the radian...

a)Prove that the function
u(x, y) = x -y÷x+y
is harmonic and obtain a conjugate function v(x, y) such that
f(z) = u + iv is analytic.
b)Convert the integral
from 0 to 5 of (25-t²)^3/2 dt
into a Beta Function and evaluate the resulting function.
c)Solve the first order PDE
sin(x) sin(y)
∂u
∂x + cos(x) cos(y)
∂u
∂y = 0
such that u(x, y) = cos(2x), on x + y =
π
2

1. Use the given conditions to find the exact value of the
expression.
sin(α) = -5/3, tan(α) > 0, sin(α - 5π/3)
2. Use the given conditions to find the exact value of the
expression.
cos α = 24/25, sin α < 0, cos(α + π/6)
3. Use the given conditions to find the exact value of the
expression.
cot x = √3, cos x < 0, tan(x + π/6)
4. If α and β are acute angles such that...

Evaluate the following.
f(x, y) = x + y
S: r(u, v) = 5
cos(u) i + 5 sin(u)
j + v k, 0 ≤ u
≤ π/2, 0 ≤ v ≤ 3

Identify the surface with parametrization x = 3 cos θ sin φ, y =
3 sin θ sin φ, z = cos φ where 0 ≤ θ ≤ 2π and 0 ≤ φ ≤ π. Hint: Find
an equation of the form F(x, y, z) = 0 for this surface by
eliminating θ and φ from the equations above. (b) Calculate a
parametrization for the tangent plane to the surface at (θ, φ) =
(π/3, π/4).

13. Consider
f(x)=sqrt x-2
a) Using any of the three limit formulas to find f ′ ( a
), what is the slope of the tangent line to f ( x )at x = 18? (6
points execution, 2 points notation)
b) Find the equation of the tangent line at x =
18
14. State the derivative.
a) d/ d x [ x ^n ]
b) d /d x [ cos x ]
c) d /d x [ csc...

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