Chapter 4: Problem 3
Find the indefinite integral. $$ \int \frac{1}{3-2 x} d x $$
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Find any relative extrema of the function. Use a graphing utility to confirm your result. \(g(x)=x \operatorname{sech} x\)
Find the limit. \(\lim _{x \rightarrow \infty} \tanh x\)
Consider a particle moving along the \(x\) -axis where \(x(t)\) is the position of the particle at time \(t, x^{\prime}(t)\) is its velocity, and \(\int_{a}^{b}\left|x^{\prime}(t)\right| d t\) is the distance the particle travels in the interval of time. A particle moves along the \(x\) -axis with velocity \(v(t)=1 / \sqrt{t}\) \(t > 0\). At time \(t=1,\) its position is \(x=4\). Find the total distance traveled by the particle on the interval \(1 \leq t \leq 4\).
Verify the differentiation formula. \(\frac{d}{d x}[\operatorname{sech} x]=-\operatorname{sech} x \tanh x\)
In Exercises 87-89, consider a particle moving along the \(x\) -axis where \(x(t)\) is the position of the particle at time \(t, x^{\prime}(t)\) is its velocity, and \(\int_{a}^{b}\left|x^{\prime}(t)\right| d t\) is the distance the particle travels in the interval of time. The position function is given by \(x(t)=t^{3}-6 t^{2}+9 t-2\) \(0 \leq t \leq 5 .\) Find the total distance the particle travels in 5 units of time.
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