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- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/03%3A_Determinants/3.02%3A_Properties_of_DeterminantsThere are many important properties of determinants. Since many of these properties involve the row operations discussed in Chapter 1, we recall that definition now. We will now consider the effect of...There are many important properties of determinants. Since many of these properties involve the row operations discussed in Chapter 1, we recall that definition now. We will now consider the effect of row operations on the determinant of a matrix. In future sections, we will see that using the following properties can greatly assist in finding determinants. This section will use the theorems as motivation to provide various examples of the usefulness of the properties.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/03%3A_Determinants/3.02%3A_Properties_of_Determinants/3.2E%3A_Exercises_for_Section_3.2This page includes exercises on matrix operations, specifically focusing on determinants. It explains how row and column operations affect determinants, discusses properties linked to nilpotent and or...This page includes exercises on matrix operations, specifically focusing on determinants. It explains how row and column operations affect determinants, discusses properties linked to nilpotent and orthogonal matrices, and provides proofs regarding matrix similarities that maintain determinant values.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Discrete_Mathematics_for_Computer_Science_(Fitch)/01%3A_Basics/1.01%3A_TerminologyThis page discusses the structure of mathematics, emphasizing the importance of proofs and fundamental categories like undefined terms, defined terms, axioms, and theorems. Undefined terms prevent inf...This page discusses the structure of mathematics, emphasizing the importance of proofs and fundamental categories like undefined terms, defined terms, axioms, and theorems. Undefined terms prevent infinite definitions, while defined terms rely on these and established definitions. Axioms are unproven statements serving as a foundation for deriving new statements. Mastering these concepts is essential for solving mathematical problems.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_I%3A_Differential_Calculus/01%3A_Functions_and_Graphs/1.06%3A_Chapter_1_Review_ExercisesThis page outlines mathematical exercises focused on functions, relations, and various equations, including trigonometric and logarithmic problems. It involves tasks such as evaluating statements abou...This page outlines mathematical exercises focused on functions, relations, and various equations, including trigonometric and logarithmic problems. It involves tasks such as evaluating statements about functions, determining domains and ranges, and solving equations. The page also covers practical applications, like T-shirt production costs and population modeling in Ocean City, along with carbon dating methods to estimate the age of a skeleton through radiocarbon decay.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Discrete_Mathematics_for_Computer_Science_(Fitch)/02%3A_Logic/2.04%3A_Mathematical_ProofThis page discusses the concept of proof in mathematics, emphasizing its role in understanding and communication. It covers two theorems about sets: one states that the intersection of two sets is a s...This page discusses the concept of proof in mathematics, emphasizing its role in understanding and communication. It covers two theorems about sets: one states that the intersection of two sets is a subset of either, and the other indicates that the set difference is part of the complement of the second set. Both theorems include detailed proofs, along with practice checkpoints for further investigation.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/01%3A_Integration/1.08%3A_Chapter_1_Review_ExercisesThis page features calculus exercises on definite integrals, Riemann sums, and antiderivatives. It includes exercises on evaluating mathematical truths and real-world applications, such as calculating...This page features calculus exercises on definite integrals, Riemann sums, and antiderivatives. It includes exercises on evaluating mathematical truths and real-world applications, such as calculating average costs and velocities. The content ranges from theoretical proofs to practical scenarios, emphasizing the continuity of functions and derivatives. Specific calculations and their answers are provided, demonstrating the connections between theory and application.
- 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).