1. Write down a concise, closed-form formula for the derivative of each of the following functions. \[ f(x) = x^{10} + 10^x + \tfrac{10}{x} + \tfrac{x}{10} + \sqrt[10]{x} + \sqrt[x]{10} + 10^{10} \qquad g(x) = \frac{\ln(x)}{\sin(x)\cos(x)} \qquad p(t) = \sin\Bigl(2^{\ln(t)}\Bigr) \qquad q(x) = \log_{8}\bigl(x^2 + 10x + 25\bigr) \]
  2. Consider the graph of the function \(f(x) = \cos\bigl(\mathrm{e}^x+1\bigr).\) What’s an equation for the line secant to the graph of \(f\) between the points where \(x = 1\) and \(x = 3?\) What’s an equation for the line tangent to the graph of \(f\) at the point where \(x = 2?\) Express each of these linear equations in the form \(y = mx+b\) with parameters \(m\) and \(b\) accurate to within \(\pm 0.001.\)

  3. Imagine you’re standing at the rooftop of St Mary’s Medical Center which, at 148 ft high, is the tallest building in Grand Junction. From the rooftop’s edge you hurl a glass bottle of barbecue sauce downwards towards the parking lot below with an initial velocity of 37 ft/s. The height of the bottle from the parking lot \(t\) seconds after you throw it can be modelled by the function \(h(t) = 148-37t-16t^2.\)

    (a) How long after you throw it does the bottle hit the ground?
    (b) What is the average speed of the bottle between the moment you throw it and the moment it hits the ground?
    (c) What is the instantaneous speed of the bottle the moment it hits the ground?

  4. Suppose you invest $1,234 in a money-market savings account that offers a \(2.5\%\) annual interest rate compounded monthly.

    (a) By what percent does the balance of your account increase after the first year?
    (b) What is the instantaneous rate-of-change of you account balance, expressed in dollars-per-year, at the three year mark?
    (c) What is that same instantaneous rate-of-change at the three year mark expressed as a percent of your current balance? I.e. what’s the force of interest for your money-market account?