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Questions and Answers
What is the first step to compute $99(2x - 198)^2 x - 99$ dx?
What is the first step to compute $99(2x - 198)^2 x - 99$ dx?
What is the volume of a square pyramid with base area $r^2$ and height $h$?
What is the volume of a square pyramid with base area $r^2$ and height $h$?
If $f$ is an integrable function on $[0,1]$, which statement is true about $|f|$?
If $f$ is an integrable function on $[0,1]$, which statement is true about $|f|$?
What does the well-ordering principle state?
What does the well-ordering principle state?
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If $ ext{lim}{x o p^+} f(x) = ext{lim}{x o p^-} f(x) = A$, what can be concluded about $ ext{lim}_{x o p} f(x)$?
If $ ext{lim}{x o p^+} f(x) = ext{lim}{x o p^-} f(x) = A$, what can be concluded about $ ext{lim}_{x o p} f(x)$?
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To find $(f^{-1})'(0)$ where $f(x) = rac{1}{2} ext{cos}( ext{sin} t)$, what must be known?
To find $(f^{-1})'(0)$ where $f(x) = rac{1}{2} ext{cos}( ext{sin} t)$, what must be known?
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What can be concluded if a function $f(x)$ is continuous on $[0, 1]$ and $f(0) = f(1)$?
What can be concluded if a function $f(x)$ is continuous on $[0, 1]$ and $f(0) = f(1)$?
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Which of the following properties is true for the function defined on $[-1, 1]$ that is continuous and differentiable but has a derivative that is not continuous?
Which of the following properties is true for the function defined on $[-1, 1]$ that is continuous and differentiable but has a derivative that is not continuous?
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Study Notes
Practice Exam 1
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Problem 1: Calculate the definite integral $\int_{}^{} (2x - 198)^{2} [\sqrt{x} - 99] dx$. The function includes the greatest integer function.
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Problem 2: Find the volume of the square pyramid using Cavalieri's theorem. The pyramid has a base area of $r^2$ and height $h$.
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Problem 3: Prove that the absolute value $|f|$ of an integrable function $f$ on the interval [0,1] is also integrable.
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Problem 4: Show that the well-ordering principle implies the principle of mathematical induction. This involves a set $S$ of natural numbers, and a subset $T$.
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Problem 5: If the limit of a function $f(x)$ as $x$ approaches $p$ from the left and right both equal $A$, prove that the limit as $x$ approaches $p$ is also $A$.
Practice Exam 2
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Problem 1: Evaluate the limit $\lim_{h\to0} \frac{\int_{0}^{1+h} e^{t^2} dt - \int_{0}^{1} e^{t^2} dt}{h(3+h^2)}$, using a theorem.
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Problem 2: Find the derivative of the inverse function $(f^{-1})'(0)$, for a specific function $f(x)$ involving a definite integral.
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Problem 3: Find the value of $f(2)$ given a definite integral equation. The function $f(x)$ is continuous.
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Problem 4: Provide an example of a function $f(x)$ on the interval [-1,1] that is continuous and differentiable but whose derivative $f'(x)$ is not continuous at certain points.
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Problem 5: Demonstrate that if $f(x)$ is continuous on [0,1] and $f(0)=f(1)$, then there exists a point $x\in[0,1]$ such that $f(x)=f(x+1/n)$ for any positive integer $n$.
Practice Exam 3
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Problem 1: Evaluate the integral $\int_{}^{} \frac{t + t^2}{\sqrt{1 + t^2}} dt$
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Problem 2: Calculate the integral $\int_{}^{} x^2 \sqrt{x^2 - 9} dx$
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Problem 3: Given $\lim_{x→a^+} g(x) = B$ (a finite, non-zero value) and $\lim_{x→a^+} h(x) = 0$, with $h(x) \neq 0$ near $a$, prove that $\lim_{x→a^+} \frac{g(x)}{h(x)}$ is infinity.
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Problem 4: If $f(x)$ is a positive continuous function on [0,$\infty$) with $\lim_{x→∞} f(x) = 0$, prove that there exists a maximum value $M$ for $f(x)$ on the interval [0, $\infty$).
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Problem 5: Demonstrate that a convergent sequence is a Cauchy sequence. A function $f$ is a contraction if there exists 0 < a < 1 such that $|f(x) - f(y)| ≤ a|x-y|$. Show that if $f$ is a contraction in the composition of $f$ on itself $n$ times, the sequence formed is Cauchy.
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Description
This quiz covers various topics in calculus and analysis including definite integrals, volume calculations using Cavalieri's theorem, and principles of mathematical induction. It features complex problem-solving and proofs related to limits and integrability. Perfect for those preparing for advanced mathematics exams.
