Calculus Examples

$f (x) = x^{3} - 2$ , $(1, - 1)$

Step 1

Check if the given point is on the graph of the given function.

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Step 1.1

Evaluate $f (x) = x^{3} - 2$ at $x = 1$ .

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Step 1.1.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = {(1)}^{3} - 2$

Step 1.1.2

Simplify the result.

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Step 1.1.2.1

One to any power is one.

$f (1) = 1 - 2$

Step 1.1.2.2

Subtract $2$ from $1$ .

$f (1) = - 1$

Step 1.1.2.3

The final answer is $- 1$ .

$- 1$

Step 1.2

Since $- 1 = - 1$ , the point is on the graph.

The point is on the graph

Step 2

The slope of the tangent line is the derivative of the expression.

$m$ $=$ The derivative of $f (x) = x^{3} - 2$

Step 3

Consider the limit definition of the derivative.

$f' (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

Step 4

Find the components of the definition.

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Step 4.1

Evaluate the function at $x = x + h$ .

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Step 4.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = {(x + h)}^{3} - 2$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Use the Binomial Theorem.

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 2$

Step 4.1.2.2

The final answer is $x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 2$ .

$x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 2$

Step 4.2

Reorder.

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Step 4.2.1

Move $x^{2}$ .

$x^{3} + 3 h x^{2} + 3 x h^{2} + h^{3} - 2$

Step 4.2.2

Move $x$ .

$x^{3} + 3 h x^{2} + 3 h^{2} x + h^{3} - 2$

Step 4.2.3

Move $x^{3}$ .

$3 h x^{2} + 3 h^{2} x + h^{3} + x^{3} - 2$

Step 4.2.4

Move $3 h x^{2}$ .

$3 h^{2} x + h^{3} + 3 h x^{2} + x^{3} - 2$

Step 4.2.5

Reorder $3 h^{2} x$ and $h^{3}$ .

$h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} - 2$

Step 4.3

Find the components of the definition.

$f (x + h) = h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} - 2$

$f (x) = x^{3} - 2$

$f (x + h) = h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} - 2$

$f (x) = x^{3} - 2$

Step 5

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} - 2 - (x^{3} - 2)}{h}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Apply the distributive property.

$f' (x) = lim_{h \to 0} \frac{h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} - 2 - x^{3} + 2}{h}$

Step 6.1.2

Multiply $- 1$ by $- 2$ .

$f' (x) = lim_{h \to 0} \frac{h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} - 2 - x^{3} + 2}{h}$

Step 6.1.3

Subtract $x^{3}$ from $x^{3}$ .

$f' (x) = lim_{h \to 0} \frac{h^{3} + 3 h^{2} x + 3 h x^{2} + 0 - 2 + 2}{h}$

Step 6.1.4

Add $h^{3}$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{h^{3} + 3 h^{2} x + 3 h x^{2} - 2 + 2}{h}$

Step 6.1.5

Add $- 2$ and $2$ .

$f' (x) = lim_{h \to 0} \frac{h^{3} + 3 h^{2} x + 3 h x^{2} + 0}{h}$

Step 6.1.6

Add $h^{3} + 3 h^{2} x + 3 h x^{2}$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{h^{3} + 3 h^{2} x + 3 h x^{2}}{h}$

Step 6.1.7

Factor $h$ out of $h^{3} + 3 h^{2} x + 3 h x^{2}$ .

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Step 6.1.7.1

Factor $h$ out of $h^{3}$ .

$f' (x) = lim_{h \to 0} \frac{h \cdot h^{2} + 3 h^{2} x + 3 h x^{2}}{h}$

Step 6.1.7.2

Factor $h$ out of $3 h^{2} x$ .

$f' (x) = lim_{h \to 0} \frac{h (h^{2}) + h (3 h x) + 3 h x^{2}}{h}$

Step 6.1.7.3

Factor $h$ out of $3 h x^{2}$ .

$f' (x) = lim_{h \to 0} \frac{h (h^{2}) + h (3 h x) + h (3 x^{2})}{h}$

Step 6.1.7.4

Factor $h$ out of $h (h^{2}) + h (3 h x)$ .

$f' (x) = lim_{h \to 0} \frac{h (h^{2} + 3 h x) + h (3 x^{2})}{h}$

Step 6.1.7.5

Factor $h$ out of $h (h^{2} + 3 h x) + h (3 x^{2})$ .

$f' (x) = lim_{h \to 0} \frac{h (h^{2} + 3 h x + 3 x^{2})}{h}$

Step 6.2

Reduce the expression by cancelling the common factors.

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Step 6.2.1

Cancel the common factor of $h$ .

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Step 6.2.1.1

Cancel the common factor.

$f' (x) = lim_{h \to 0} \frac{h (h^{2} + 3 h x + 3 x^{2})}{h}$

Step 6.2.1.2

Divide $h^{2} + 3 h x + 3 x^{2}$ by $1$ .

$f' (x) = lim_{h \to 0} h^{2} + 3 h x + 3 x^{2}$

Step 6.2.2

Simplify the expression.

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Step 6.2.2.1

Move $h$ .

$f' (x) = lim_{h \to 0} h^{2} + 3 x h + 3 x^{2}$

Step 6.2.2.2

Move $h^{2}$ .

$f' (x) = lim_{h \to 0} 3 x h + 3 x^{2} + h^{2}$

Step 6.2.2.3

Reorder $3 x h$ and $3 x^{2}$ .

$f' (x) = lim_{h \to 0} 3 x^{2} + 3 x h + h^{2}$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$lim_{h \to 0} 3 x^{2} + lim_{h \to 0} 3 x h + lim_{h \to 0} h^{2}$

Step 8

Evaluate the limit of $3 x^{2}$ which is constant as $h$ approaches $0$ .

$3 x^{2} + lim_{h \to 0} 3 x h + lim_{h \to 0} h^{2}$

Step 9

Move the term $3 x$ outside of the limit because it is constant with respect to $h$ .

$3 x^{2} + 3 x lim_{h \to 0} h + lim_{h \to 0} h^{2}$

Step 10

Move the exponent $2$ from $h^{2}$ outside the limit using the Limits Power Rule.

$3 x^{2} + 3 x lim_{h \to 0} h + {(lim_{h \to 0} h)}^{2}$

Step 11

Evaluate the limits by plugging in $0$ for all occurrences of $h$ .

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Step 11.1

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$3 x^{2} + 3 x \cdot 0 + {(lim_{h \to 0} h)}^{2}$

Step 11.2

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$3 x^{2} + 3 x \cdot 0 + 0^{2}$

Step 12

Simplify the answer.

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Step 12.1

Simplify each term.

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Step 12.1.1

Multiply $3 x \cdot 0$ .

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Step 12.1.1.1

Multiply $0$ by $3$ .

$3 x^{2} + 0 x + 0^{2}$

Step 12.1.1.2

Multiply $0$ by $x$ .

$3 x^{2} + 0 + 0^{2}$

Step 12.1.2

Raising $0$ to any positive power yields $0$ .

$3 x^{2} + 0 + 0$

Step 12.2

Combine the opposite terms in $3 x^{2} + 0 + 0$ .

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Step 12.2.1

Add $3 x^{2}$ and $0$ .

$3 x^{2} + 0$

Step 12.2.2

Add $3 x^{2}$ and $0$ .

$3 x^{2}$

Step 13

Find the slope $m$ . In this case $m = 3$ .

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Step 13.1

Remove parentheses.

$m = 3 \cdot 1^{2}$

Step 13.2

Simplify $3 \cdot 1^{2}$ .

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Step 13.2.1

One to any power is one.

$m = 3 \cdot 1$

Step 13.2.2

Multiply $3$ by $1$ .

$m = 3$

Step 14

The slope is $m = 3$ and the point is $(1, - 1)$ .

$m = 3, (1, - 1)$

Step 15

Find the value of $b$ using the formula for the equation of a line.

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Step 15.1

Use the formula for the equation of a line to find $b$ .

$y = m x + b$

Step 15.2

Substitute the value of $m$ into the equation.

$y = (3) \cdot x + b$

Step 15.3

Substitute the value of $x$ into the equation.

$y = (3) \cdot (1) + b$

Step 15.4

Substitute the value of $y$ into the equation.

$- 1 = (3) \cdot (1) + b$

Step 15.5

Find the value of $b$ .

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Step 15.5.1

Rewrite the equation as $(3) \cdot (1) + b = - 1$ .

$(3) \cdot (1) + b = - 1$

Step 15.5.2

Multiply $3$ by $1$ .

$3 + b = - 1$

Step 15.5.3

Move all terms not containing $b$ to the right side of the equation.

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Step 15.5.3.1

Subtract $3$ from both sides of the equation.

$b = - 1 - 3$

Step 15.5.3.2

Subtract $3$ from $- 1$ .

$b = - 4$

Step 16

Now that the values of $m$ (slope) and $b$ (y-intercept) are known, substitute them into $y = m x + b$ to find the equation of the line.

$y = 3 x - 4$

Step 17