Calculus

Derivatives

Discrete First Derivatives Warmup

         

n123456an=n2149162536Δ(an)=an+1an35??? \begin{array} {c|c|c|c|c|c|c|c} n & 1 & 2 & 3 & 4 & 5 & 6 & \cdots \\ a_n = n^2 & \color{#D61F06}{1} & \color{#D61F06}{4} & \color{#D61F06}{9} & 16 & 25 & 36 & \cdots \\ \Delta(a_n) = a_{n+1} - a_n & \color{#3D99F6}{3} & \color{#3D99F6}{5} & ???\\ \end{array}

The second row of the table above shows the first few terms of the sequence an=n2.a_n = n^2. Each entry in the third row is the difference between the next term in row 2 and the current term in row 2. For example, 3=41\color{#3D99F6}{3} = \color{#D61F06}{4} -\color{#D61F06}{1} and 5=94.\color{#3D99F6}{5} = \color{#D61F06}{9} - \color{#D61F06}{4}. What number should go in the cell marked '???'?

n123456an=n3182764125216Δ(an)=an+1an7193761??? \begin{array} {c|c|c|c|c|c|c|c} n & 1 & 2 & 3 & 4 & 5 & 6 & \cdots \\ a_n = n^3 & 1 & 8 & 27 & 64 & 125 & 216 & \cdots \\ \Delta(a_n) = a_{n+1} - a_n & 7 & 19 & 37 & 61 & ???\\ \end{array}

The second row of the table above shows the first few terms of the sequence an=n3.a_n = n^3. Each entry in the third row is the difference between the next term in row 2 and the current term in row 2. For example, 7=817 = 8 -1 and 19=278.19 = 27 -8. What number should go in the cell marked '???'?

n123456nn+1an=n3182764125216n3(n+1)3Δ(an)=an+1an719376191? \begin{array} {c|c|c|c|c|c|c|c} n & 1 & 2 & 3 & 4 & 5 & 6 & \cdots & n & n +1 \\ a_n = n^3 & 1 & 8 & 27 & 64 & 125 & 216 & \cdots &n^3 & (n+1)^3 \\ \Delta(a_n) = a_{n+1} - a_n & 7 & 19 & 37 & 61 & 91 & & \cdots & ? \\ \end{array}

Given an=n3,a_n = n^3, find Δ(an)\Delta(a_n).

In other words, what is the general formula for the discrete derivative of {n3}?\{n^3\}?

Given an=n2+n,a_n = n^2 + n, find Δ(an),\Delta(a_n), the discrete derivative of {an}.\{a_n\}.

Which of these sequences has a constant discrete derivative?

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