Find the Tangent at a Given Point Using the Limit Definition f(x)=x^2 , (0,0)

$f (x) = x^{2}$ , $(0, 0)$

Step 1

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

Evaluate $f (x) = x^{2}$ at $x = 0$ .

Replace the variable $x$ with $0$ in the expression.

$f (0) = {(0)}^{2}$

Simplify the result.

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

$f (0) = 0$

The final answer is $0$ .

$0$

Since $0 = 0$ , 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 $f (x) = x^{2}$

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.

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

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

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

Simplify the result.

Rewrite ${(x + h)}^{2}$ as $(x + h) (x + h)$ .

$f (x + h) = (x + h) (x + h)$

Expand $(x + h) (x + h)$ using the FOIL Method.

Apply the distributive property.

$f (x + h) = x (x + h) + h (x + h)$

Apply the distributive property.

$f (x + h) = x \cdot x + x h + h (x + h)$

Apply the distributive property.

$f (x + h) = x \cdot x + x h + h x + h \cdot h$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x + h) = x^{2} + x h + h x + h \cdot h$

Multiply $h$ by $h$ .

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

Add $x h$ and $h x$ .

Reorder $x$ and $h$ .

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

Add $h x$ and $h x$ .

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

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

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

Reorder.

Move $x^{2}$ .

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

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

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

Find the components of the definition.

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

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

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

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

Step 5

Plug in the components.

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

Step 6

Simplify.

Simplify the numerator.

Subtract $x^{2}$ from $x^{2}$ .

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

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

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

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

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

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

Factor $h$ out of $2 h x$ .

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

Factor $h$ out of $h (h) + h (2 x)$ .

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

Reduce the expression by cancelling the common factors.

Cancel the common factor of $h$ .

Cancel the common factor.

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

Divide $h + 2 x$ by $1$ .

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

Reorder $h$ and $2 x$ .

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

Step 7

Evaluate the limit.

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

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

Evaluate the limit of $2 x$ which is constant as $h$ approaches $0$ .

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

Step 8

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

$2 x + 0$

Step 9

Add $2 x$ and $0$ .

$2 x$

Step 10

Multiply $2$ by $0$ .

$m = 0$

Step 11

The slope is $m = 0$ and the point is $(0, 0)$ .

$m = 0, (0, 0)$

Step 12

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

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

$y = m x + b$

Substitute the value of $m$ into the equation.

$y = (0) \cdot x + b$

Substitute the value of $x$ into the equation.

$y = (0) \cdot (0) + b$

Substitute the value of $y$ into the equation.

$0 = (0) \cdot (0) + b$

Find the value of $b$ .

Rewrite the equation as $(0) \cdot (0) + b = 0$ .

$(0) \cdot (0) + b = 0$

Simplify $(0) \cdot (0) + b$ .

Multiply $0$ by $0$ .

$0 + b = 0$

Add $0$ and $b$ .

$b = 0$

Step 13

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.

$y = 0$

Step 14

Find the Tangent at a Given Point Using the Limit Definition f(x)=x^2 , (0,0)

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