Download Summer Calculus BC Homework

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Summer Calculus BC Homework
Part 2
Topics from Pre-Calculus H
Period:
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It is presumed that every student in Calculus BC is coming from successful completion of either PreCalculus Honors, or Calculus AB. The following questions are from Pre-Calculus Honors and
demonstrate algebra or trigonometric skills that we will need to call upon in Calculus BC. Even if you
have never completed Pre-Calculus Honors (e.g. you took Pre-Calculus CP then Calculus AB), you will
still be expected to perform all of these (and other) algebra and trigonometry tasks that students saw in
Pre-Calculus Honors. At the end of each item is a short statement for Calculus AB students, about where
in Calculus we will see these skills.
 Power Reduction Formulas. Which of the following is equivalent to
(A) 2(1  cos[ 32 x])
(B)
(D) 1  cos[6 x]
(E)
Use in Calculus:
 cos  d
2
1
(1  cos[ 23 x])
2
1
(1  cos[6 x])
4
1
2
sin 2 (3 x) ?
(C) 2(1  cos[6 x])
is not readily integrable. But use of a power reduction formula
makes this expression possible to integrate.
 Inverse Trigonometry. Which of the following is an algebraic expression for sin(tan 1 x) ?
(A)
x
x 2 1
(B)
x
x 2 1
(C)
1
x 2 1
(D)
1
1 x 2
(E)
x 2 1
x
Use in Calculus: one of our methods of integration is called “trigonometric substitutions” and we
use inverse trigonometry in order to go back and forth between algebraic and
trigonometric expressions.
 Polar Graphing: Which of the following curves represents the polar coordinate graph of r 2  sin 2 ?
(A)
(B)
(D)
(E)
(C)
Use in Calculus: we will be computing the slopes of lines tangent to polar curves, the area
enclosed by polar curves, and the arc lengths of polar curves. In short—
everything you can do in Calculus in ( x, y ) coordinates we can also do in
polar ( r ,  ) coordinates as well.
More polar: [Calculator active] The polar curve r is given by r ( )  3  sin  , where 0    2 .
For 2     , there is one point P on the curve with x-coordinate 3 . Find the angle  that
corresponds to P and find the y-coordinate of P. Show the equations and set ups you used to lead to
your answers and then use a graphing calculator to solve those equations to three decimal places.
More polar: Shown at right are the graphs of r1  2sin  and r2  2cos  .
1. State the points of intersection of the two curves on the
interval [0,  ] .
2. The shaded region represents the area between the two
curves for some angles,      . Give the values of 
and of  .
y
3. Is the shaded region symmetric about the ray   4 ?
2
 Parametric Graphing: Use your graphing
calculator in parametric mode to sketch the
graph of the curve defined parametrically by:
1
–2
3t
1  t3
.
3t 2
y (t ) 
1  t3
x (t ) 
–1
1
2
–1
–2
Use in Calculus: many things in the world are not easily modeled by simple y  f ( x) functions
but rather by parametric functions. We can still calculate tangent lines, arc
lengths, concavity, and volumes of regions of revolution… But it is all in
parametric now too.
 Hyperbolic Trigonometric Functions: What is cosh(0) ?
(A) 0
(B)
1
e
(C) ln 2
(D) 1
(E)
e
2
Use in Calculus: just as we can differentiate and integrate all of the trigonometric functions—
sine, cosine, tangent, secant, cosecant, and cotangent, we can also differentiate
and integrate all of the hyperbolic trigonometric functions—cosh, sinh, tanh,
sech, csch, and coth.
x
More Hyperbolic Trigonometry: What is lim tanh x ?
x 
(A) 0
(B) 1
(C)

2
(D) e
(E) does not exist
Use in Calculus: The hyperbolic trigonometric functions are not only used to model various
phenomena (such as logistical growth, or catenary curves) but for hyperbolic
trig substitutions. Students completing Calculus AB should recall an integral
dx
formula such as
 tan 1 x  C . It turns out that we will also learn
2
x 1
dx
things like
 tanh 1 x  C in Calculus BC.
1  x2


 Vectors: Consider the vector given by v(t )  t 2  2t , 2  t . When t  3 , what is v(t ) ?
(A) 2
(B)
10
(C) 4
(D) 9
(E) 16
Use in Calculus: particle motion is often described parametrically and that can in turn be
described with vectors. Students should recall that speed is the magnitude of
velocity, so if a particle’s velocity can be described by a vector, we will need to
get the magnitude of that vector to get the particle’s speed.
[These next skills were actually introduced way back in Algebra 2 Honors, but they are still applicable to
Calculus and we should be able to do them. Plus, we used them again in Pre-Calculus Honors.]
 Long Division: Which of the following is equivalent to
3x
1
 2
x 1 x 1
5x
1
(D) 2 x  2  2
 2
x 1 x 1
(A) 2 x  2 
2
2 x3  3x  1
?
x2  1
x
1
 2
x 1 x 1
1
(E) 2 x  1  2
x 1
(B) 2 x 
2
(C) 2 x 
3
1

x  1 x2  1
Use in Calculus: we often times need to perform long division on rational polynomials to antidifferentiate them. A student of Calculus should realize that the first step in
2 x3  3x  1
dx is to do long division to break up the fraction into much
x2  1
more integrable parts.

 Partial Fraction Decomposition: What is the partial fraction decomposition of
5 3

x3 x
3 2x 1

(D)
x x2  5
(A)
5 x  1 14

5x
x3
3
29
(E) 5  
x x5
(B)
(C)
5 x 2  x  15
?
x3  5 x
3
1
 2
x x 5
Another Partial Fraction Decomposition: What is the correct format for the decomposition of
p ( x)
? (note that the numerator is left unspecified as p( x) so that you cannot find
( x  2)2 ( x2  4)
numeric values for this decomposition).
Ax  B Cx  D

( x  2)2 ( x2  4)
A
Bx
Cx  D
(D)

 2
2
( x  2) ( x  2)
( x  4)
A
B
 2
2
( x  2)
( x  4)
A
Bx  C
(C)
 2
2
( x  2)
( x  4)
A
B
Cx  D
(E)

 2
2
( x  2) ( x  4)
( x  2)
(A)
(B)
Use in Calculus: this skill is frequently combined with the previous one. One of our major
techniques of integration is to start with a partial fraction decomposition.
 Factoring Out Negative Exponents: Which of the following is equivalent to
(A)
(D)
1  2x
23
3x (1  x)2
1  13 x 2 3
3x 2 3 (1  x)2
(B)
1  x  x 1 3
3 x 2 3 (1  x) 2
(E)
x 1 3  x 2 3  x1 3
3 x 2 3 (1  x) 2
(C)
1
3
x 2 3 (1  x)  (1) x1 3
(1  x)2
?
3  2x
3x (1  x)2
23
Use in Calculus: it turns out that the expression in the beginning of the problem is the derivative
x1 3
(using the quotient rule) of y 
. We generally want to find when a
1 x
derivative is equal to zero or undefined, and that is far easier to do by
1 2 3
x (1  x)  (1) x1 3
simplifying 3
first.
(1  x)2
 Summation: Which of the following gives the value of 4  16  36  64  100  144 
 4n 2 ? (Note:
n
the above sum can also be expressed as
 4i
2
in “sigma” notation.)
i 1
(A)  n(n  1)
2
(D)
2n(n  1)(2n  1)
3
n(n  1)(2n  1)
6
2(n  1)(n  2)(n  3)
(E)
3
(B)
(C)
n(2n  1)(2n  3)
2
Use in Calculus: we will need to use the laws of summation and the formulas for certain well
known sums in order to compute “Riemann Sums” that in essence give the area
of arbitrarily bound regions in the xy plane.