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 \( \frac{d}{d[]} ( [] ) |_{x=[]} \).
As you complete the tasks below, what patterns do you notice?
Constants: Let \( f(x) = \) ____, where ____ is a different constant chosen by each group member.
\(f'(1) = \) _____ \(= 0\)
\(f'(2) = \) _____ \(= 0\)
\(f'(3) = \) _____ \(= 0\)
\(f'(4) = \) _____ \(= 0\)
\(f'(5) = \) _____ \(= 0\)
Squares: Let \( f(x) = x^2 \). After computing results, try to rewrite each answer showing a common factor. For example, 6 and 10 have a common factor of 2 so could be written as: \(2 \cdot 3\) and \(2 \cdot 5\).
\(f'(1) = \) _____ \(= 2 \cdot 1\)
\(f'(2) = \) _____ \(= 2 \cdot 2\)
\(f'(3) = \) _____ \(= 2 \cdot 3\)
\(f'(4) = \) _____ \(= 2 \cdot 4\)
\(f'(5) = \) _____ \(= 2 \cdot 5\)
Cubics: Let \( f(x) = x^3 \). If possible, rewrite each answer showing a common factor.
\(f'(1) = \) _____ \(= 3 \cdot 1\)
\(f'(2) = \) _____ \(= 3 \cdot 4\)
\(f'(3) = \) _____ \(= 3 \cdot 9\)
\(f'(4) = \) _____ \(= 3 \cdot 16\)
\(f'(5) = \) _____ \(= 3 \cdot 25\)
Quartics: Let \( f(x) = x^4 \). If possible, rewrite each answer showing a common factor.
\(f'(1) = \) _____ \(= 4 \cdot 1\)
\(f'(2) = \) _____ \(= 4 \cdot 8\)
\(f'(3) = \) _____ \(= 4 \cdot 27\)
\(f'(4) = \) _____ \(= 4 \cdot 64\)
\(f'(5) = \) _____ \(= 4 \cdot 125\)
The derivative "power rule":
Coefficients: Let \( f(x) = c \cdot x^3 \). The group should agree on a coefficient \( c \) (please do not use 0 or 1). (Hint: It may help to compare to "cubes" above.) Assume \(c = -1\)
\(f'(1) = \) _____ \(= -3 \cdot 1\)
\(f'(2) = \) _____ \(= -3 \cdot 4\)
\(f'(3) = \) _____ \(= -3 \cdot 9\)
\(f'(4) = \) _____ \(= -3 \cdot 16\)
\(f'(5) = \) _____ \(= -3 \cdot 25\)
\(e^x\): Let \( g(x) = e^x \). Compute \( g(x) \) and \( g'(x) \).
\(g(1) = \) _____; \(g'(1) = \) _____ both \(2.718\ldots\)
\(g(2) = \) _____; \(g'(2) = \) _____ both \(7.389\ldots\)
\(g(3) = \) _____; \(g'(3) = \) _____ both \(20.0855\ldots\)
\(g(4) = \) _____; \(g'(4) = \) _____ both \(53.598\ldots\)
\(g(5) = \) _____; \(g'(5) = \) _____ both \(148.413\ldots\)
\(\ln(x)\): Let \(h(x) = \ln(x)\). Compute \( h'(x) \). (Hint: It may help to use fraction form.)
\(h'(1) = \) _____ \(= 1\)
\(h'(2) = \) _____ \(= \dfrac{1}{2}\)
\(h'(3) = \) _____ \(= \dfrac{1}{3}\)
\(h'(4) = \) _____ \(= \dfrac{1}{4}\)
\(h'(5) = \) _____ \(= \dfrac{1}{5}\)
Challenge
On the same graph, plot \( x^2 \) and \( \frac{d}{dx} (x^2) \) using nDeriv(). How did you do this?