Brilliance Math Tutoring

💡 Something pretty cool about math is how we can determine a value, t, within a particular interval of a continuous, differentiable function, that represents a tangent line that is parallel to a line connecting the endpoints defined by the interval. Simply put, there is a value, t, that satisfies the following condition.

s’(t) = [s(b) - s(a)] / (b - a)
Where “a” and “b” are the lower and upper ends of the given interval.

In essence, we are working backwards to satisfy arbitrary conditions.

✔ Step 1: Verify continuity. The function exists at every point in the interval.
✔ Also step 1: Verify differentiability. Does the given function even have a derivative?
✔ Step 2: Find the average rate of change.
✔ Step 3: Calculate the derivative of the given function.
✔ Step 4: Apply the today’s theorem to find t.

📝 The opposite of last week's discussion topic, derivations, are also fun and involved in its own unique ways. Depending on how the given function presents itself (whether a product of two functions, quotient of two functions, a sum, difference, or a composition), the process can easily become intricate. Today's examples examine a few of these instances.

🤔 Try this! Identify the error(s) in the derivation of v(w) in part a).