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Calculus

Derivatives

Concept Quizzes

Derivatives Warmup
Derivative by First Principle
Derivatives of Linear Functions
Derivatives of Polynomials
Derivatives of Rational Function
Derivatives of Exponential Functions
Derivatives of Logarithmic Functions
Derivatives of Trigonometric Functions
Instantaneous Rate of Change
Graphical Interpretation of Derivatives
Discrete First Derivatives Warmup

Challenge Quizzes

Derivatives: Level 2 Challenges

Discrete First Derivatives Warmup

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

The second row of the table above shows the first few terms of the sequence $a_n = n^2.$ 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, $\color{#3D99F6}{3} = \color{#D61F06}{4} -\color{#D61F06}{1}$ and $\color{#3D99F6}{5} = \color{#D61F06}{9} - \color{#D61F06}{4}.$ What number should go in the cell marked '???'?

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by Brilliant Staff

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

The second row of the table above shows the first few terms of the sequence $a_n = n^3.$ 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, $7 = 8 -1$ and $19 = 27 -8.$ What number should go in the cell marked '???'?

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by Brilliant Staff

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

Given $a_n = n^3,$ find $\Delta(a_n)$ .

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

3

3n^2

3n^2+3n + 1

3n^2+1

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by Brilliant Staff

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

2n

2n+1

2n + 2

n^3

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by Brilliant Staff

Which of these sequences has a constant discrete derivative?

a_n =5n + 3

a_n = n^3

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by Brilliant Staff