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- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/04%3A_Parametric_EquationsIn this chapter, we also study parametric equations, which give us a convenient way to describe curves or to study the position of a particle or object in two dimensions as a function of time. We will...In this chapter, we also study parametric equations, which give us a convenient way to describe curves or to study the position of a particle or object in two dimensions as a function of time. We will use parametric equations to describe many topics later in this text.
- https://math.libretexts.org/Courses/De_Anza_College/Math_1D%3A_De_Anza/02%3A_Multiple_Integration/2.02%3A_Double_Integrals_over_General_Regions/2.2E%3A_ExercisesThis page covers the classification of regions in calculus as Type I and Type II for integral evaluations, detailing calculations for areas and volumes under specified functions. It includes practical...This page covers the classification of regions in calculus as Type I and Type II for integral evaluations, detailing calculations for areas and volumes under specified functions. It includes practical exercises on double integrals and explores geometric interpretations in three-dimensional space. Additionally, it discusses the lunes of Alhazen, proving their area is equivalent to that of a corresponding triangle.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/02%3A_Applications_of_First_Order_Equations/2.05%3A_Orthogonal_Trajectories_of_CurvesThis page discusses orthogonal trajectories of a family of curves through differential equations, emphasizing the condition for perpendicular intersection through derivatives. It presents an example i...This page discusses orthogonal trajectories of a family of curves through differential equations, emphasizing the condition for perpendicular intersection through derivatives. It presents an example involving parabolas and demonstrates how to find their orthogonal trajectories, resulting in ellipses. The solution process includes parameter elimination and solving a differential equation, with a figure depicting the parabolas and their corresponding orthogonal ellipses.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/02%3A_Applications_of_Integration/2.01%3A_Areas_between_Curves/2.1E%3A_Exercises_for_Section_2.1This page provides exercises on determining areas between curves through integration, covering various function types and methods. It includes analyses of different pairs of curves, calculations of ar...This page provides exercises on determining areas between curves through integration, covering various function types and methods. It includes analyses of different pairs of curves, calculations of area, and practical applications like marginal costs and revenues. The text features geometric representations and specific examples, including a business scenario related to profits and a comparison of speed functions in a race.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/02%3A_Applications_of_Integration/2.01%3A_Areas_between_CurvesJust as definite integrals can be used to find the area under a curve, they can also be used to find the area between two curves. To find the area between two curves defined by functions, integrate th...Just as definite integrals can be used to find the area under a curve, they can also be used to find the area between two curves. To find the area between two curves defined by functions, integrate the difference of the functions. If the graphs of the functions cross, or if the region is complex, use the absolute value of the difference of the functions. In this case, it may be necessary to evaluate two or more integrals.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_I%3A_Differential_Calculus/05%3A_Differential_Calculus_with_Parametric_Curves/5.03%3A_Chapter_5_Review_ExercisesSketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. x=1+t,y=t2−1,−1≤t≤1 x=et,y=1−e3t,0≤t≤1 For, \(x=\ln(t),\; y=t^2−...Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. x=1+t,y=t2−1,−1≤t≤1 x=et,y=1−e3t,0≤t≤1 For, x=ln(t),y=t2−1,t=1, find the equation of the tangent line to the given curve. Find dydx,dxdy, and d2xdy2 of y=(2+e−t),x=1−sint dydx=1etcost,dxdy=etcost,d2xdy2=e2t(sint−cost).
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/02%3A_Applications_of_First_Order_Equations/2.05%3A_Orthogonal_Trajectories_of_Curves/2.5E%3A_Exercises_for_Section_2.5This page presents exercises aimed at finding orthogonal trajectory curves for specific mathematical curves like y2=ax3 and y=aex. Each exercise challenges the reader to identify the co...This page presents exercises aimed at finding orthogonal trajectory curves for specific mathematical curves like y2=ax3 and y=aex. Each exercise challenges the reader to identify the corresponding orthogonal curves, with some offering answers such as y2+2x=C or (y−x)3(y+x)=k, while others require the derivation of solutions. The exercises encompass various forms, including polynomial and exponential curves.
