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Problem 1

Estimate the limits numerically. \(\lim _{x \rightarrow 0} \frac{x^{2}}{x+1}\)

Expert verified

The limit as x approaches 0 of the function \(\frac{x^2}{x+1}\) is 0, which we can verify analytically and confirm numerically as the function approaches 0 for values very close to 0: \[
\lim_{x \rightarrow 0} \frac{x^2}{x+1} = 0
\]

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Chapter 10

Compute the indicated derivative. $$ U(t)=5.1 t^{2}+5.1 ; U^{\prime}(3) $$

Chapter 10

Explain why we cannot put \(h=0\) in the approximation $$ f^{\prime}(x) \approx \frac{f(x+h)-f(x)}{h} $$ for the derivative of \(f\).

Chapter 10

Compute the derivative function \(f^{\prime}(x)\) algebraically. (Notice that the functions are the same as those in Exercises \(1-14 .)\) HINT [See Examples 2 and \(3 .]\) $$ f(x)=2 x-x^{2} $$

Chapter 10

Having been soundly defeated in the national lacrosse championships, Brakpan High has been faced with decreasing sales of its team paraphernalia. However, sales, while still decreasing, appear to be bottoming out. What does this tell you about the derivative of the sales curve?

Chapter 10

Sketch the graph of a function whose derivative exceeds 1 at every point

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