Chapter 4: Problem 60
Find an equation for the function \(f\) that has the indicated derivative and whose graph passes through the given point. $$ f^{\prime}(x)=x^{2} e^{-0.2 x^{3}}, \quad\left(0, \frac{3}{2}\right) $$
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Consider the function \(F(x)=\frac{1}{2} \int_{x}^{x+2} \frac{2}{t^{2}+1} d t\) (a) Write a short paragraph giving a geometric interpretation of the function \(F(x)\) relative to the function \(f(x)=\frac{2}{x^{2}+1}\) Use what you have written to guess the value of \(x\) that will make \(F\) maximum. (b) Perform the specified integration to find an alternative form of \(F(x)\). Use calculus to locate the value of \(x\) that will make \(F\) maximum and compare the result with your guess in part (a).
Show that if \(f\) is continuous on the entire real number line, then \(\int_{a}^{b} f(x+h) d x=\int_{a+h}^{b+h} f(x) d x\)
Find the limit. \(\lim _{x \rightarrow \infty} \tanh x\)
Evaluate the integral in terms of (a) natural logarithms and (b) inverse hyperbolic functions. \(\int_{-1 / 2}^{1 / 2} \frac{d x}{1-x^{2}}\)
(a) integrate to find \(F\) as a function of \(x\) and (b) demonstrate the Second Fundamental Theorem of Calculus by differentiating the result in part (a). $$ F(x)=\int_{\pi / 3}^{x} \sec t \tan t d t $$
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