💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!

Hi all! My name is Tyler Moulton, and I'm a recent graduate of Harvard University where I studied Applied Mathematics with a focus field of Astronomy and a secondary in Planetary Sciences. I'm passionate about education and research, with my long-term career goal being to become a Professor of Astronomy and Planetary Sciences. I spent two of my summers at Harvard tutoring students, and also served as a teaching fellow for Ordinary and Partial Differential Equations in the Applied Math Department. I'm excited to bring this experience into my work as a Numerade Educator and provide top notch mathematics instruction to everyone!

Explain how each graph is obtained from the graph of $ y = f(x) $.

(a) $ y = f(x) + 8 $(b) $ y = f (x + 8) $(c) $ y = 8f(x) $(d) $ y = f(8x) $(e) $ y = -f(x) - 1 $(f) $ y = 8f (\frac{1}{8}x) $

Find the numbers at which $ f $ is discontinuous. At which of these numbers is $ f $ continuous from the right, from the left, or neither? Sketch the graph of $ f $.

$ f(x) = \left\{ \begin{array}{ll} x^2 & \mbox{if $ x < -1 $}\\ x & \mbox{if $ -1 \le x < 1 $} \\ 1/x & \mbox{if $ x \ge 1 $} \end{array} \right.$

Each limit represents the derivative of some function $ f $ at some number $ a $. State such an $ f $ and $ a $ in each case.

$ \displaystyle \lim_{\theta \to \pi/6} \frac{\sin \theta - \frac{1}{2}}{\theta - \pi/6} $

The cost (in dollars) of producing $ x $ units of a certain commodity is $ C(x) = 5000 + 10x + 0.05x^2 $.

(a) Find the average rate of change of $ C $ with respect to $ x $ when the production level is changed (i) from $ x = 100 $ to $ x = 105 $ (ii) from $ x = 100 $ to $ x = 101 $

(b) Find the instantaneous rate of change of $ C $ with respect to $ x $ when $ x = 100 $. ( This is called the \textit{marginal cost}. Its significance will be explained in Section 3.7.)

The graph shows the influence of the temperature $ T $ on the maximum sustainable swimming speed $ S $ of Coho salmon.

(a) What is the meaning of the derivative $ S'(T) $? What are its units?

(b) Estimate the values of $ S'(15) $ and $ S'(25) $ and interpret them.

The figure shows a circular arc of length $ s $ and $ a $ chord of length $ d, $ both subtended by a central angle $ \theta $. Find$ \displaystyle \lim_{\theta \to 0+} \frac {s}{d} $