You CAN Ace Calculus

### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

### Search Practice Problems

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On this page, we discuss hyperbolas in rectangular coordinates. For a discussion of hyperbolas in polar form, see this separate page.

 Hyperbola

The hyperbola is the most complicated of the three and, consequently, the most interesting. Figure 1 contains information about the hyperbola.
The vertices ($$\pm a$$ on this plot) and the foci (F1 and F2 on this plot) lie on the transverse axis with the center at $$(h,k)$$. The standard equations are

 $$\displaystyle{ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 }$$ horizontal transverse axis $$\displaystyle{ \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 }$$ vertical transverse axis

As we did with the ellipse, we define an intermediate value $$c^2=a^2+b^2$$ which will help us locate the foci.
For a hyperbola, we need to know the equations of the lines in blue on the plot. These are asymptotes.
For a horizontal transverse axis, the asymptotes are $$\displaystyle{ y=k \pm \frac{b}{a}(x-h) }$$.
For a vertical transverse axis, the asymptotes are $$\displaystyle{ y=k \pm \frac{a}{b}(x-h) }$$.
Similar to the ellipse, we define the eccentricity as $$e=c/a$$. The results are summarized next.

$$\displaystyle{ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 }$$

horizontal transverse axis

center

$$(h,k)$$

vertices

$$(h \pm a, k)$$

foci

$$(h \pm c,k)$$

asymptotes

$$\displaystyle{ y=k \pm \frac{b}{a}(x-h) }$$

$$c^2=a^2+b^2$$

eccentricity

$$e=c/a$$

$$\displaystyle{ \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 }$$

vertical transverse axis

center

$$(h,k)$$

vertices

$$(h, k \pm a)$$

foci

$$(h,k \pm c)$$

asymptotes

$$\displaystyle{ y=k \pm \frac{a}{b}(x-h) }$$

$$c^2=a^2+b^2$$

eccentricity

$$e=c/a$$

 Notes

1. Notice that the value of c is different here than for an ellipse.
2. Since $$c > a$$, $$e > 1$$.

The hyperbola is quite a complicated graph with lots of features. Figure 2 gives you an idea of some of the other features that are involved with hyperbolas. Here is a video that goes into more detail.

### MIP4U - Conic Sections: The Hyperbola part 1 of 2 [4min-12secs]

video by MIP4U

Okay, time for some practice problems on hyperbolas. Although it may seem like we have given you a lot of information on this page, we have just skimmed the surface in the discussion of these three figures, especially the hyperbola. There are a lot more interesting features that we hope you get to explore in your class.

### Practice

Sketch the graph of the hyperbola $$\displaystyle{ (y+4)^2 - \frac{x^2}{25} = 1 }$$.

Problem Statement

Sketch the graph of the hyperbola $$\displaystyle{ (y+4)^2 - \frac{x^2}{25} = 1 }$$.

Solution

### 1592 solution video

video by PatrickJMT

Graph $$\displaystyle{ \frac{x^2}{49} - \frac{y^2}{25} = 1 }$$.

Problem Statement

Graph $$\displaystyle{ \frac{x^2}{49} - \frac{y^2}{25} = 1 }$$.

Solution

### 1604 solution video

video by PatrickJMT

Write the equation of the hyperbola that has vertices $$(-2,-5), (4,-5)$$ and foci $$(-4,-5), (6,-5)$$.

Problem Statement

Write the equation of the hyperbola that has vertices $$(-2,-5), (4,-5)$$ and foci $$(-4,-5), (6,-5)$$.

Solution

### 1605 solution video

video by PatrickJMT

Graph $$\displaystyle{ \frac{(x-2)^2}{4} - \frac{(y+3)^2}{9} = 1 }$$.

Problem Statement

Graph $$\displaystyle{ \frac{(x-2)^2}{4} - \frac{(y+3)^2}{9} = 1 }$$.

Solution

### 1606 solution video

video by MIP4U

Graph $$\displaystyle{ \frac{(y+4)^2}{4} - \frac{(x-2)^2}{16} = 1 }$$.

Problem Statement

Graph $$\displaystyle{ \frac{(y+4)^2}{4} - \frac{(x-2)^2}{16} = 1 }$$.

Solution

video by MIP4U