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  • 4.1: Intervals
    https://math.libretexts.org/Bookshelves/Analysis/A_Primer_of_Real_Analysis_(Sloughter)/04%3A_Topology_of_the_Real_Line/4.01%3A_Intervals
    (a,b)={x:x∈R,a<x<b} [a,b]={x:x∈R,a≤x≤b}, (−∞,b]={x:x∈R,x≤b}, [a,+∞)={x:x∈R,x≥a} \[(...(a,b)={x:x∈R,a<x<b} [a,b]={x:x∈R,a≤x≤b}, (−∞,b]={x:x∈R,x≤b}, [a,+∞)={x:x∈R,x≥a} (a,b]={x:x∈R,a<x≤b} [a,b)={x:x∈R,a≤x<b}, (a,b)={x:x=λa+(1−λ)b,0<λ<1}. x=(b−xb−a)a+(x−ab−a)b=(b−xb−a)a+(1−b−xb−a)b
  • 6.1: Best Linear Approximations
    https://math.libretexts.org/Bookshelves/Analysis/A_Primer_of_Real_Analysis_(Sloughter)/06%3A_Derivatives/6.01%3A_Best_Linear_Approximations
    We say f is differentiable at a if there exists a linear function dfa:R→R such that limx→af(x)−f(a)−dfa(x−a)x−a=0. We ...We say f is differentiable at a if there exists a linear function dfa:R→R such that limx→af(x)−f(a)−dfa(x−a)x−a=0. We call the function dfa the best linear approximation to f at a, or the differential of f at a.
  • 1.2: Functions
    https://math.libretexts.org/Bookshelves/Analysis/A_Primer_of_Real_Analysis_(Sloughter)/01%3A_Fundamentals/1.02%3A_Functions
    If A and B are sets, we call a relation R⊂A×B a function with domain A if for every a∈A there exists one, and only one, b∈B such that (a,b)∈R. ...If A and B are sets, we call a relation R⊂A×B a function with domain A if for every a∈A there exists one, and only one, b∈B such that (a,b)∈R. We typically indicate such a relation with the notation f:A→B, and write f(a)=b to indicate that (a,b)∈R. We call the set of all b∈B such that f(a)=b for some a∈A the range of f. With this notation, we often refer to R as the graph of \(f\…
  • 1.3: Rational Numbers
    https://math.libretexts.org/Bookshelves/Analysis/A_Primer_of_Real_Analysis_(Sloughter)/01%3A_Fundamentals/1.03%3A_Rational_Numbers
    Note that if (p,q)∈P, then (−p,q)∼(p,−q). Hence, if a=pq∈Q, then we let −a=−pq=p−q. For any a,b∈Q, we will write \(...Note that if (p,q)∈P, then (−p,q)∼(p,−q). Hence, if a=pq∈Q, then we let −a=−pq=p−q. For any a,b∈Q, we will write a−b to denote a+(−b). If a=pq∈Q with p≠0, then we let a−1=qp. Moreover, we will write 1a=a−1, 1an=a−n for any n∈Z+, and, for any b∈Q, ba=ba−1. …
  • Glossary
    https://math.libretexts.org/Bookshelves/Calculus/The_Calculus_of_Functions_of_Several_Variables_(Sloughter)/zz%3A_Back_Matter/20%3A_Glossary
    Example and Directions Words (or words that have the same definition) The definition is case sensitive (Optional) Image to display with the definition [Not displayed in Glossary, only in pop-up on pag...Example and Directions Words (or words that have the same definition) The definition is case sensitive (Optional) Image to display with the definition [Not displayed in Glossary, only in pop-up on pages] (Optional) Caption for Image (Optional) External or Internal Link (Optional) Source for Definition "Genetic, Hereditary, DNA ...") (Eg. "Relating to genes or heredity") The infamous double helix CC-BY-SA; Delmar Larsen Glossary Entries Definition Image Sample Word 1 Sample Definition 1
  • 2.4: The Fundamental Theorem of Integrals
    https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Yet_Another_Calculus_Text__A_Short_Introduction_with_Infinitesimals_(Sloughter)/02%3A_Integrals/2.04%3A_The_Fundamental_Theorem_of_Integrals
    The main theorem of this section is key to understanding the importance of definite integrals. In particular, we will invoke it in developing new applications for definite integrals. Moreover, we will...The main theorem of this section is key to understanding the importance of definite integrals. In particular, we will invoke it in developing new applications for definite integrals. Moreover, we will use it to verify the fundamental theorem of calculus.
  • 2.2: Best Affine Approximations
    https://math.libretexts.org/Bookshelves/Calculus/The_Calculus_of_Functions_of_Several_Variables_(Sloughter)/02%3A_Functions_from_R_to_R/2.02%3A_Best_Affine_Approximations
    &=\left(f_{1}^{\prime}(t), f_{2}^{\prime}(t), \ldots, f_{n}^{\prime}(t)\right)+\left(g_{1}^{\prime}(t), g_{2}^{\prime}(t), \ldots, g_{n}^{\prime}(t)\right) \nonumber \\ \frac{d}{d t}(f(t) \cdot g(t))=...&=\left(f_{1}^{\prime}(t), f_{2}^{\prime}(t), \ldots, f_{n}^{\prime}(t)\right)+\left(g_{1}^{\prime}(t), g_{2}^{\prime}(t), \ldots, g_{n}^{\prime}(t)\right) \nonumber \\ \frac{d}{d t}(f(t) \cdot g(t))=& \frac{d}{d t}\left(f_{1}(t) g_{1}(t)+f_{2}(t) g_{2}(t)+\cdots+f_{n}(t) g_{n}(t)\right) \nonumber \\ =& f_{1}(t) g_{1}^{\prime}(t)+f_{1}^{\prime}(t) g_{1}(t)+f_{2}(t) g_{2}^{\prime}(t)+f_{2}^{\prime}(t) g_{2}(t)+\cdots \label{} \\
  • Back Matter
    https://math.libretexts.org/Bookshelves/Calculus/The_Calculus_of_Functions_of_Several_Variables_(Sloughter)/zz%3A_Back_Matter
  • 3.6.E: Definite Integrals (Exercises)
    https://math.libretexts.org/Bookshelves/Calculus/The_Calculus_of_Functions_of_Several_Variables_(Sloughter)/03%3A_Functions_from_R_to_R/3.06%3A_Definite_Integrals/3.6.E%3A_Definite_Integrals_(Exercises)
    Evaluate ∬, where D is the region bounded by the x-axis, the y-axis, and D the line y=2-x. (d) \(\int_{0}^{1} \int_{0}^{x} \int_{0}^{x+y} \int_{0}^{x+y+z} w...Evaluate \iint_{D} x y d x d y, where D is the region bounded by the x-axis, the y-axis, and D the line y=2-x. (d) \int_{0}^{1} \int_{0}^{x} \int_{0}^{x+y} \int_{0}^{x+y+z} w d w d z d y d x Evaluate \iiint_{D} x y d x d y d z, where D is the region bounded by the xy-plane, the yz-plane, the xz-plane, and the plane with equation z = 4 − x − y.
  • 6.5: L'Hopital's Rule
    https://math.libretexts.org/Bookshelves/Analysis/A_Primer_of_Real_Analysis_(Sloughter)/06%3A_Derivatives/6.05%3A_L'Hopital's_Rule
    \lim _{x \rightarrow a^{+}} \frac{f^{\prime}(x)}{g^{\prime}(x)}=\lambda . whenever x \in(a, a+\delta) . Now, by the Generalized Mean Value Theorem, for any x and y with \(a<x<y<a+\delt...\lim _{x \rightarrow a^{+}} \frac{f^{\prime}(x)}{g^{\prime}(x)}=\lambda . whenever x \in(a, a+\delta) . Now, by the Generalized Mean Value Theorem, for any x and y with a<x<y<a+\delta, there exists a point c \in(x, y) such that Suppose a, b \in \mathbb{R}, f and g are differentiable on (a, b), g^{\prime}(x) \neq 0 for all x \in(a, b), and \lim _{x \rightarrow b^{-}} \frac{f^{\prime}(x)}{g^{\prime}(x)}=\lambda . \nonumber
  • 4.4.E: Green's Theorem (Exercises)
    https://math.libretexts.org/Bookshelves/Calculus/The_Calculus_of_Functions_of_Several_Variables_(Sloughter)/04%3A_Functions_from_R'_to_R/4.04%3A_Green's_Theorem/4.4.E%3A_Green's_Theorem_(Exercises)
    (a) \int_{\partial D} 2 x y d x+3 x^{2} d y (a) \int_{\partial D} 2 x y d x+3 x^{2} d y=80 (a) \int_{\partial D} 2 x y^{2} d x+4 x d y (b) \int_{\partial D} y d x+x d y (a) \(\int_{\pa...(a) \int_{\partial D} 2 x y d x+3 x^{2} d y (a) \int_{\partial D} 2 x y d x+3 x^{2} d y=80 (a) \int_{\partial D} 2 x y^{2} d x+4 x d y (b) \int_{\partial D} y d x+x d y (a) \int_{\partial D} 2 x y^{2} d x+4 x d y=\frac{16}{3} \text { (c) } \int_{\partial D} y d x-x d y=-8 \int_{\partial D} p d x+q d y=0 , \nonumber \iint_{D}\left(\frac{\partial q}{\partial x}-\frac{\partial p}{\partial y}\right) d x d y=\int_{C_{1}} p d x+q d y+\int_{C_{2}} p d x+q d y , \nonumber

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