Calculator lab - nDeriv

Today we are exploring what graphing calculators can do with derivatives.

TI-84 command: MATH → nDeriv()

Recall our work with \( f(x) = x^2 \). Try f'(1) = nDeriv(x^2, x, 1) or \( \frac{d}{d[]} ( [] ) |_{x=[]} \).

For each of the situations below, compute \( f'(a) \) for \( a \in \{ 1, 2, 3, 4, 5 \} \). What patterns do you notice?

  1. Constants: Let \( f(x) = c \), where \( c \) is a different constant choosen by each group member.
  2. Squares: Let \( f(x) = x^2 \). If possible, write each \( f'(a) \) showing a common factor. For example, 6 and 10 have a common factor of 2 so could be written as: \(2 \cdot 3\) and \(2 \cdot 5\).
  3. Cubics: Let \( f(x) = x^3 \). If possible, write each \( f'(a) \) showing a common factor.
  4. Quartics: Let \( f(x) = x^4 \). If possible, write each \( f'(a) \) showing a common factor.
  5. Coefficients:Let \( f(x) = c \cdot x^2 \). The group should agree on a constant \( c \) (please do not use 0 or 1). (Hint: It may help to compare to "squares" above.)
  6. e: Let \( g(x) = e^x \) . Compute \( g(a) \) and \( g'(a) \).
  7. Natural log:Let \( h(x) = \ln(x) \). Compute \( h'(a) \). (Hint: It may help to use fraction form.)

Challenge 1

Use the definition of a derivative to verify the pattern for \( f(x) = x^3 \).


Challenge 2

On the same graph, plot \( x^2 \) and \( \frac{d}{dx} (x^2) \) using nDeriv(). How did you do this?