Home
/ Ap Calculus Bc Vectors in the Plane Review Solutions Pdf
Ap Calculus Bc Vectors in the Plane Review Solutions Pdf
AP Calculus BC Review: Vector-Valued Functions
I thing that sets the AP Calculus BC exam autonomously from the AB examination is the topic of vector-valued functions. The BC test has them, while the AB does not. In this article we will review how to graph, discover derivatives and integrals, and interpret the meaning of vector-valued functions.
What are Vector-Valued Functions?
A vector-valued function is like a typical part y = f(x), except that there is more than one output value. In fact, a vector may be thought of every bit a list of multiple values, such as (i, iv, -2).
On the AP Calculus BC exam, you will only encounter vector-valued functions having two outputs. Y'all volition see these as a pair of functions, x = f(t) and y = k(t), or equivalently, (f(t), g(t)).
For instance, (3 cos t, 5 sin t) is a vector-valued function. It specifies the x-value (iii cos t) and y-value (5 sin t) for any given input t.
The graph of a vector-valued function, x=3cos(t), y = 5sin(t).
When there is a single input variable (like t in the higher up example), then a vector-valued role is essentially the aforementioned thing as a set of parametric equations.
Both of these concepts are basic types of multivariable functions. Check out AP Calculus Review: Multivariables for more than.
Graphing
Let's take a closer look at the case given to a higher place, (3 cos t, 5 sin t). How do yous graph it?
Well what I did to create the graph shown above was to use graphing software.
In fact, near graphing calculators are capable of graphing vector functions. Nevertheless, this feature will most likely exist under term parametric. Hither is a expert parametric graphing tutorial.
Even so what do yous do on the parts of the exam that exercise non let a calculator?
Sketching the Graph Without a Calculator
Most often you will not need a detailed graph to answer any particular question. However, it's good to have a few techniques upwards your sleeve to help you lot visualize a vector office when you need to.
Some situations come up upward and then oftentimes that y'all should probably memorize them.
For positive constants a and b, (a cos t, b sin t) is an ellipse. If a = b, then information technology's a circumvolve of radius r = a.
Suppose f is any office. Then (t, f(t)) produces exactly the same graph every bit y = f(10). The simply deviation is that (t, f(t)) is a vector function, and then all the methods that brand sense for vector functions would work on this one.
More than mostly, if you need to know something about the graph of (f(t), g(t)), then yous could choose a few sample points for t and plot the corresponding (x, y) pairs.
For example, to sketch (t2 – 6, t – ii) by hand, showtime create a table of values. Then plot the points and connect with a smooth curve.
t
x = t2 - vi
y = t - 2
Point
-3
3
-5
(3, -5)
-2
-2
-4
(-two, -iv)
-1
-five
-3
(-5, -iii)
0
-vi
-2
(-half dozen, -2)
one
-five
-ane
(-5, -1)
2
-two
0
(-two, 0)
3
3
1
(3, 1)
This graph is a parabola opening sideways to the correct. The arrowhead on the curve indicates the direction of increasing t.
Derivatives
Only as you can take derivatives of regular (not-vector) functions, y'all tin have derivatives of vector functions too. In fact, it's like shooting fish in a barrel! Just accept the derivative of each component role separately, and keep your answer in the form of a vector.
For case, the derivative of the vector function (t2 – 6, t – 2) is (2t, 1).
In the next section, we'll run across what the derivative can tell you about the motion of an object described by a vector-valued function.
Position, Velocity, and Acceleration
Remember that if s(t) is the position part for an object moving along a straight line, then you can discover the object'southward velocity and acceleration by taking derivatives.
In a similar mode, if the vector office (f(t), g(t)) defines the position of an object in terms of time t, then its velocity and acceleration (as vector functions) tin can also be plant by taking derivatives:
In any case, whether talking nigh single-variable functions or vector-valued functions, the velocity involves the first derivative while the acceleration vector involves the 2nd.
Example – Velocity Vector
If a particle moves in a the xy-plane and then that at time t > 0 its position vector is , so its velocity vector at time t = 3 is
A. (ln vi, ln 27)
B. (ln 9, ln 27)
C. (6e9, -27e-27)
D. (9e9, -27e-27)
Solution
From what we have already discussed, velocity is the outset derivative of position. Therefore, existence very careful with the Chain Rule, we find:
And so later on plugging in the given time t = 3, nosotros become:
The correct choice is C.
Length of a Bend and Distance
Although we typically recall of parametric curves when talking nigh curve length, the same formula works just as well for vector-valued functions.
In fact, the length formula gives you the total distance of a particle traveling along a vector function (f(t), m(t)) from time t = a to t = b.
Example – Distance
A particle moves along a path described past . Find the distance that the particle travels along the path from t = 0 to t = π/2. (Presume that calculators are immune.)
Solution
This is a straightforward awarding of the length integral. Use your reckoner once you have set up the formula correctly.
Summary
Vector-valued functions are an of import role of the AP Calculus BC exam. You lot should know how to:
Graph and interpret the graph of a vector-valued office.
Find derivatives and use them in particle motion problems.
Utilize the curve length formula.
For a complete list of topics found on the BC exam, check out What Topics are on the AP Calculus BC Test?.
Shaun earned his Ph. D. in mathematics from The Ohio State University in 2008 (Get Bucks!!). He received his BA in Mathematics with a pocket-size in information science from Oberlin College in 2002. In add-on, Shaun earned a B. Mus. from the Oberlin Conservatory in the aforementioned year, with a major in music composition. Shaun all the same loves music -- most equally much as math! -- and he (thinks he) tin can play piano, guitar, and bass. Shaun has taught and tutored students in mathematics for near a decade, and hopes his experience can help y'all to succeed!
View all posts
By the way, Magoosh can help y'all written report for both the Sat and ACT exams. Click here to learn more than!
