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Sazzad Sazid

Sazzad S.

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Viewed Questions

Let $p$ and $q$ be the propositions "The election is decided" and "The votes have been counted, " respectively. Express each of these compound propositions an English sentence. a) $\neg p$ b) $p \vee q$ c) $\neg p \wedge q$ d) $q \rightarrow p$ e) $\neg q \rightarrow \neg p$ f) $\neg p \rightarrow \neg q$ g) $p \leftrightarrow q$ h) $\neg q \vee(\neg p \wedge q)$

Let $p$ and $q$ be the propositions "The election is decided" and "The votes have been counted, " respectively. Express each of these compound propositions an English sentence. a) $\neg p$ b) $p \vee q$ c) $\neg p \wedge q$ d) $q \rightarrow p$ e) $\neg q \rightarrow \neg p$ f) $\neg p \rightarrow \neg q$ g) $p \leftrightarrow q$ h) $\neg q \vee(\neg p \wedge q)$

Discrete Mathematics and its Applications

The Foundations: Logic and Proofs

Propositional Logic
Show that each of these conditional statements is a tautology by using truth tables. a) $[\neg p \wedge(p \vee q)] \rightarrow q$ b) $[(p \rightarrow q) \wedge(q \rightarrow r)] \rightarrow(p \rightarrow r)$ c) $[p \wedge(p \rightarrow q)] \rightarrow q$ d) $[(p \vee q) \wedge(p \rightarrow r) \wedge(q \rightarrow r)] \rightarrow r$

Show that each of these conditional statements is a tautology by using truth tables. a) $[\neg p \wedge(p \vee q)] \rightarrow q$ b) $[(p \rightarrow q) \wedge(q \rightarrow r)] \rightarrow(p \rightarrow r)$ c) $[p \wedge(p \rightarrow q)] \rightarrow q$ d) $[(p \vee q) \wedge(p \rightarrow r) \wedge(q \rightarrow r)] \rightarrow r$

Discrete Mathematics and its Applications

The Foundations: Logic and Proofs

Propositional Equivalences
Construct a truth table for each of these compound propositions. a) $p \rightarrow(\neg q \vee r)$ b) $\neg p \rightarrow(q \rightarrow r)$ c) $(p \rightarrow q) \vee(\neg p \rightarrow r)$ d) $(p \rightarrow q) \wedge(\neg p \rightarrow r)$ e) $(p \rightarrow q) \vee(\neg q \rightarrow r)$ f) $(\neg p \leftrightarrow \neg q) \leftrightarrow(q \leftrightarrow r)$

Construct a truth table for each of these compound propositions. a) $p \rightarrow(\neg q \vee r)$ b) $\neg p \rightarrow(q \rightarrow r)$ c) $(p \rightarrow q) \vee(\neg p \rightarrow r)$ d) $(p \rightarrow q) \wedge(\neg p \rightarrow r)$ e) $(p \rightarrow q) \vee(\neg q \rightarrow r)$ f) $(\neg p \leftrightarrow \neg q) \leftrightarrow(q \leftrightarrow r)$

Discrete Mathematics and its Applications

The Foundations: Logic and Proofs

Propositional Logic

Questions asked

INSTANT ANSWER

Topic : Analysis of function 1. Consider the functions i. \( \quad y=f(x)=x^{4}-4 x^{3}+4 x^{2}=x^{2}\left(x^{2}-4 x+4\right)=x^{2}(x-2)^{2} \) ii. \( \quad y=x^{5}-x^{3}=x^{3}(x+1)(x-1) \) Write down the following information's (a) Find the intercepts of the function. (b) Find the critical, stationary and inflection points of the function. (c) Find the intervals on which \( f(x) \) is increasing or decreasing. (d) Find the interval on which \( f(x) \) is concave up and down. (e) Find the relative extremum of \( f(x) \) using both first and second derivative test. (f) Find the end behavior of \( f(x) \) (g) Make a conjecture about the behavior of the graph of \( y=f(x) \) in the vicinity of its \( x \)-intercepts and test your conjecture by generating the graph.

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INSTANT ANSWER

Topic : Analysis of function 1. Consider the functions i. \( \quad y=f(x)=x^{4}-4 x^{3}+4 x^{2}=x^{2}\left(x^{2}-4 x+4\right)=x^{2}(x-2)^{2} \) ii. \( \quad y=x^{5}-x^{3}=x^{3}(x+1)(x-1) \) Write down the following information's (a) Find the intercepts of the function. (b) Find the critical, stationary and inflection points of the function. (c) Find the intervals on which \( f(x) \) is increasing or decreasing. (d) Find the interval on which \( f(x) \) is concave up and down. (e) Find the relative extremum of \( f(x) \) using both first and second derivative test. (f) Find the end behavior of \( f(x) \) (g) Make a conjecture about the behavior of the graph of \( y=f(x) \) in the vicinity of its \( x \)-intercepts and test your conjecture by generating the graph.

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INSTANT ANSWER

10. \begin{tabular}{lcc} \hline INTERVAL & SIGN OF \( f^{\prime}(x) \) & SIGN OF \( f^{\prime \prime}(x) \) \\ \hline\( x<1 \) & + & + \\ \( 1<x<3 \) & + & - \\ \( 3<x \) & + & + \\ \hline \end{tabular}

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Use a suitable method to evaluate the area of the following shaded regions: (i) (ii)

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INSTANT ANSWER

Use a suitable method to evaluate the area of the following shaded regions: (i) (ii)

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INSTANT ANSWER

Use a suitable method to evaluate the area of the following shaded regions: (i) (ii) Evaluate the integral: \( \int_{0}^{\pi} 2 t \sin 2 t d t \)

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INSTANT ANSWER

(a) Consider, the function \( f(x)=x^{2}-2 \). (i) Find the slope of tangent line to the graph of \( f(x) \) at the point \( x=0 \). (ii) Find the equation tangent line to the graph of \( f(x) \) at the point \( x=0 \). (iii) Draw the graph of \( \boldsymbol{f}(\boldsymbol{x}) \) together with the tangent line from (ii) in the same axes.

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INSTANT ANSWER

a) The following figure represents a position function of a particle at time \( t \). (i) Find the average velocity over the time interval \( [2,6] \). (ii) What is the instantaneous velocity at \( t=8 \). Explain the reason. (iii) Determine where whe velocity doesn't exist. Explain the reason. (iv) Roughly sketch the velocity graph of the particle.

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INSTANT ANSWER

(b) Find and Sketch the area of the region enclosed by the parabola \( y=x^{2} \) and the line \( y=x+2 \). [4]

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INSTANT ANSWER

(b) The graph of \( \boldsymbol{f}(\boldsymbol{x}) \) is shown. Evaluate the following definite integrals. (i) \( \int_{-2}^{6} f(x) d x \) (ii) \( \int_{0}^{4}|f(x)| d x \) [3]

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