AP Precalculus
Resources to accompany an AP Precalculus Course
Tabla de Contenidos
- Polynomials
- Polynomial Inequality
- Rate of Change & Concavity
- Average Rate of Change
- Concavity and Average Rate of Change
- Concavity and Tangent Lines
- Rational Functions
- p.308 Example 4
- p.309 Example 6
- p.312 Example 8
- Rational Function with a Hole
- Trigonometry
- Unit Circle Trigonometry
- Unit Circle: Complementary Angles
- Unit Circle: Sine and Cosine Functions
- Graphing Sine/Cosine with Translations
- Unit Circle: The Tangent Function
- Tangent for Acute Angles
- Graphing the Tangent Function
- Lighthouse Function p.553 #69
- Parity of Trig Functions
- Reciprocal Functions
- Inverses of Trigonometric Functions
- Cosine of the Difference of Two Angles
- The Ambiguous Case of SSA
- Complex Numbers
- Polar Form of Complex Numbers
- Multiplying Complex Numbers in Polar Form
- Nth Roots of a Complex Number
- DeMoivre's Theorem: Visualization
- Polar Coordinates
- Polar Coordinates
- Polar: Distance Formula
- Circle Centered at (r₀,θ₀) with Radius d
- Tangent Line to a Circle
- Graph Polar Functions
- Polar Transformations of a Point
- Graphing Limaçons
- Graphing Lemniscates
- Rose Curves
- Lemniscate Envelope
- Vectors
- Dot Product of Two Vectors
- Dot Product: Component Form
- Projection of a Vector
- Linear Combination of Vectors
- Conic Sections
- Drawing a Parabola
- Locus Definition of a Parabola
- Parabola Properties
- Ellipse: Definition
- Ellipse: Graphing
- Hyperbola - Graphing
- Parametric Equations
- Graphing Parametric Equations
- Combining Parametric Equations
- Cycloid: Derivation
- Cycloid: Adjusting the Radius
- Cycloid Demonstration
- Epicycloid
- Deriving the Epicycloid Equations
- Hypocycloid
- Deriving the Hypocycloid Equations
- The brachistochrone and other curves
Polynomial Inequality
As you drag the slider above, watch how the sign of each of the factors of [math]f\left(x\right)[/math] changes. Each factor changes signs exactly once, as you pass its critical value. This is why the roots are called critical values, because they are the [math]x[/math]-values at which the sign of the corresponding factors changes.[br][br]If you want to see what the numerical value of each factor is, then check the "Numbers" box.
Average Rate of Change
You can type in any function you like for [math]f\left(x\right)[/math].[br][br]The graph above shows the average rate of change from a point [math]P\left(x,y\right)[/math] to a point [math]\left(x+h,f\left(x+h\right)\right)[/math]. You can set [math]h[/math], which is the change in the [math]x[/math]-coordinate. [br][br]You can also drag point [math]P[/math] anywhere you would like along the curve.[br][br]Click and drag the grid to move around. Zoom by scrolling up/down.[br]
p.308 Example 4
Example 4 from p.309[br]Done according to the directions I gave you.
Unit Circle Trigonometry
Drag the slider for [math]\theta[/math] to adjust the angle. Notice the coordinates of point [math]P[/math] on the unit circle.
Unit Circle Definition of Trig Functions
We can extend the definitions of sine, cosine and tangent (and their reciprocals) to angles of [i]any[/i] measure, by using the unit circle.[br][br]On the unit circle, if the point [math]P[/math] has coordinates [math]\left(x,y\right)[/math], then from the right triangle we can see that, in Quadrant I:[br][math]x=\cos\theta[/math][br][math]y=\sin\theta[/math][br][math]\frac{y}{x}=\tan\theta[/math][br][br]Therefore, the point where the terminal side of [math]\theta[/math] intersects the the unit circle will have coordinates [math]P\left(\cos\theta,\sin\theta\right)[/math], assuming the angle is in standard position.[br][br]Since the signs of the coordinates depend on the quadrant, the signs of sine, cosine, and tangent will also depend on the quadrant.[br][br]We can determine the coordinates of the point on the unit circle by using [math]\theta'[/math], the [i]reference angle[/i] of [math]\theta[/math]. This is the acute angle formed between the terminal side of [math]\theta[/math] and the [math]x[/math]-axis. Completing the right triangle will get us the ratios for [math]\theta'[/math], which will be identical to the ratios for [math]\theta[/math], except that we may need to attach a negative sign depending on the quadrant.[br][br]You can drag the slider for the angle and see how the red reference angle changes and where the right triangle is that will allow us to find the coordinates of the point on the unit circle and, in turn, the ratios of [math]\theta[/math].
Polar Form of Complex Numbers
Use the sliders to adjust the radius and angle for the complex number..[br]Change the Grid to polar and add the Circle and Line to see how we are graphing. Basically, we want to first find the angle, then use the radius to either go in the direction of the angle.At the top it also shows the [math]a+bi[/math] form of the number.
Polar Coordinates
Use the sliders to adjust the radius and angle for [math]P[/math].[br]Change the Grid to polar and add the Circle and Line to see how we are graphing. Basically, we want to first find the angle, then use the radius to either go in the direction of the angle (if [math]r>0[/math]) or in the opposite direction (if [math]r<0[/math])
Dot Product of Two Vectors
If we have two vectors [math]\vec{u}[/math] and [math]\vec{v}[/math] that are in the same direction, then their [b]dot product[/b] is simply the product of their magnitudes: [math]\vec{u}\cdot\vec{v}=\left|\vec{u}\right|\left|\vec{v}\right|[/math]. To see this above, drag the head of [math]\vec{v}[/math] to make it parallel to [math]\vec{u}[/math].[br][br]If the two vectors are not in the same direction, then we can find the component of vector [math]\vec{v}[/math] that is parallel to vector [math]\vec{u}[/math], which we can call [math]\vec{w}[/math]. and take the product of the magnitudes of [math]\vec{u}[/math] and [math]\vec{w}[/math]: [math]\vec{u}\cdot\vec{v}=\left|\vec{u}\right|\left|\vec{w}\right|[/math][br][br]But how can we find [math]\left|\vec{w}\right|[/math]? If the angle between vectors [math]\vec{u}[/math] and [math]\vec{v}[/math] is [math]\theta[/math], then we can see that [math]\cos\theta=\frac{\left|\vec{w}\right|}{\left|\vec{v}\right|}[/math], so [math]\left|\vec{w}\right|=\left|\vec{v}\right|\cos\theta[/math].[br][br][size=100]Therefore, in general, we have that the [b]dot product[/b] o[/size]f [math]\vec{u}[/math] and [math]\vec{v}[/math] is:[br][math]\vec{u}\cdot\vec{v}=\left|\vec{u}\right|\left|\vec{v}\right|\cos\theta[/math][br]where [math]\theta[/math] is the angle between the two vectors [math]\vec{u}[/math] and [math]\vec{v}[/math].
Drawing a Parabola
Place Points on the graph that are equidistant from F and the given line.
Graphing Parametric Equations
Graph parametric equations by entering them in terms of [math]t[/math] above. You can set the minimum and maximum values for [math]t[/math]. Pay attention to the initial point, terminal point and direction of the parametric curve.[br]