Calculus II
Review
MTH 125
January 4, 2009
The following is a review of previous mathematics courses relevant for Math 152. If you
find the material difficult, please make arrangements with me as soon as possible, or be sure
to go the study center for extra help.
1. Write tan θ, cot θ, sec θ, csc θ in terms of sin θ and cos θ.
sin θ
θ
tan θ = cos
cot θ = cos
sec θ = cos1 θ csc θ = sin1 θ
θ
sin θ
2. Which of the following can sin2 θ cot θ sec θ be simplified to:
(A) sin θ (B) cos θ (C) cot θ (D) csc θ (E) sec θ
θ
1
sin2 θ · cos
sin θ · cos θ = sin θ
£
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3. In the interval π4 , 7π
what value(s) of x is csc(x) undefined?
4
1
csc(x) = sin(x)
we must exclude values where sin(x) = 0 which happens at multiples
of π. In this interval, x 6= π
4. Find the value(s) of x over the interval 0 ≤ x ≤ π for which sin(3x) = 1.
We must have 3x = (4n+1)π
, where n ∈ Z
2
For n = 0 → x = π6
For n = 1 → x = 5π
6
For n = 2 → x = 9π
6 which is outside the interval.
Thus, sin(3x) = 1 when x = π6 and x = 5π
6 .
5. sin2 θ =?
(A) sin(sin θ)
(B) sin θ sin θ
(C) sin(2θ)
θ
(A) t = sin
(B) t = sin1 θ
7. Find the derivatives of the following:
6. sin−1 θ = t means
d
dx [sin(x)]
d
dx [cos(x)]
= cos(x)
d
dx [tan(x)]
= sec2 (x)
d
dx [cot(x)]
= − csc2 (x)
= − sin(x)
d
dx [sec(x)]
= sec(x) tan(x)
d
dx [csc(x)]
= − csc(x) cot(x)
−1
d
(x)]
dx [sin
d
mx
]
dx [e
(C) sin t = θ
=
√ 1
1−x2
d
−1
(x)]
dx [tan
d
n
dx [x ]
= memx
1
1+x2
=
= nxn−1
d
dx [ln(x)]
d
x
dx [a ]
8. Find the derivatives of the following:
√
√
√
√
1
1
t
(a) r = e t = e t · [t 2 ]0 = 21 t− 2 e t = 2e√t
(b) g(β) = sin2 (β) cos2 (β)
g 0 (β) = 2 sin(β)[sin(β)]0 cos2 (β) + sin2 (β) · 2 cos(β)[cos(β)]0
g 0 (β) = 2 sin(β) cos3 (β) − 2 cos(β) sin3 (β)
(c) m(θ) = tan(θ) ln(1 + θ 2 )
m0 (θ) = [tan(θ)]0 (ln(1 + θ2 )) + tan(θ)[ln(1 + θ2 )]0
1
2 0
m0 (θ) = sec2 (θ) (ln(1 + θ2 )) + tan(θ) 1+θ
2 [1 + θ ]
2θ
m0 (θ) = sec2 (θ) (ln(1 + θ2 )) + tan(θ) 1+θ
2
1
=
1
x
= ax (ln(a))
sin2 (x)
(d) g(x) = 2x2 +ln(x)
g 0 (x) =
g 0 (x) =
[sin2 (x)]0 (2x2 +ln(x))−[2x2 +ln(x)]0 sin2 (x)
(2x2 +ln(x))2
2 sin(x) cos(x)(2x2 +ln(x))−(4x+ x1 ) sin2 (x)
(2x2 +ln(x))2
(e) y = tan−1 (xex )
y0 =
y0 =
y0 =
1
[xex ]0
1+(xex )2
1
([x]0 ex + x[ex ]0 )
1+(xex )2
ex (1+x)
1
(ex + xex ) = 1+(xe
x )2
1+(xex )2
(f) y = sec(ln(cos(θ 2 )))
y0
y0
y0
y0
y0
= sec(ln(cos(θ2 ))) tan(ln(cos(θ2 ))) · [ln(cos(θ2 ))]0
1
2 0
= sec(ln(cos(θ2 ))) tan(ln(cos(θ2 ))) · cos(θ
2 ) · [cos(θ )]
1
2
2 0
= sec(ln(cos(θ2 ))) tan(ln(cos(θ2 ))) · cos(θ
2 ) · (− sin(θ )) · [θ ]
1
2
= sec(ln(cos(θ2 ))) tan(ln(cos(θ2 ))) · cos(θ
2 ) · (− sin(θ )) · 2θ
= −2θ tan(θ2 ) sec(ln(cos(θ2 ))) tan(ln(cos(θ2 )))
9. If the position of a function measured in feet is modeled by s(t) = 7t5 − 20t3 + 30 where
t is measured in seconds, find the acceleration after 4 seconds.
Velocity,v(t) = s0 (t) = 35t4 − 60t2
Accelerate, a(t) = v 0 (t) = s00 (t) = 140t3 − 120t
ft
After 4 seconds, a(4) = 8480 sec
2
10. Fill in the following statements:
• If f 0 (x) > 0, then f (x) is Increasing
• If f 0 (x) < 0, then f (x) is Decreasing
• If f 00 (x) > 0, then f (x) is Concave Up
• If f 00 (x) < 0, then f (x) is Concave Down
11. Given the function f (x) = x4 − 6x2 ,
(a) State the interval(s) where f (x) is increasing.
Find the critical point(s) where f 0 (x) = 0 ½
x=0√
f 0 (x) = 4x3 − 12x = 4x(x2 − 3) = 0 →
x=± 3
Create a sign chart based on f 0 (x) Clearly, f (x) is increasing on the interval(s):
√
√
x ∈ (− 3, 0) ∪ ( 3, ∞)
2
(b) State the interval(s) where f (x) is concave down.
Find the critical point(s) where f 00 (x) = 0 ©
f 00 (x) = 12x2 − 12 = 12(x2 − 1) = 0 → x = ±1
Create a sign chart based on f 00 (x) Clearly, f (x) is increasing on the interval(s):
x ∈ (−1, 1)
The above information can clearly be seen by viewing the graph of f (x).
12. Answer the following questions based on the graph of the derivative function, G0 (x)
shown below.
(a) At what x−value(s) does G(x) achieve a relative minimum?
At x = A and x = G.
(b) At what x−value(s) does G(x) achieve a relative maximum?
At x = E
(c) At what x−value(s) does G(x) achieve a point of inflection?
At x = B, x = C, x = D, x = F
(d) Sketch a possible graph of G(x).
3
13. Find an anti-derivative for each of the following:
(a) F (x) = x3 + sin(x) − x1
We are searching a function that when differentiated yields F (x). That is, find
f (x) where f 0 (x) = F (x).
f (x) = 14 x4 − cos(x) − ln(x) + C
(b) G(x) = e2x + x13
We are searching a function that when differentiated yields G(x). That is, find
g(x) where g 0 (x) = G(x).
g(x) = 12 e2x − 2x1 2 + C
1
√1
(c) H(x) = 1+x
2 +
x
We are searching a function that when differentiated yields H(x). That is, find
h(x) where h0 (x) = H(x).
√
h(x) = tan−1 (x) + 2 x + C
4