Calculus Examples

y = x^{2} + 3 x + 34

$y = x^{2} + 3 x + 34$ ,

(0, 34)

$(0, 34)$

Step 1

Write

y = x^{2} + 3 x + 34

$y = x^{2} + 3 x + 34$ as a function.

f (x) = x^{2} + 3 x + 34

$f (x) = x^{2} + 3 x + 34$

Step 2

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

Tap for more steps...

Step 2.1

Evaluate

f (x) = x^{2} + 3 x + 34

$f (x) = x^{2} + 3 x + 34$ at

x = 0

$x = 0$ .

Tap for more steps...

Step 2.1.1

Replace the variable

x

$x$ with

0

$0$ in the expression.

f (0) = {(0)}^{2} + 3 (0) + 34

$f (0) = {(0)}^{2} + 3 (0) + 34$

Step 2.1.2

Simplify the result.

Tap for more steps...

Step 2.1.2.1

Simplify each term.

Tap for more steps...

Step 2.1.2.1.1

Raising

0

$0$ to any positive power yields

0

$0$ .

f (0) = 0 + 3 (0) + 34

$f (0) = 0 + 3 (0) + 34$

Step 2.1.2.1.2

Multiply

3

$3$ by

0

$0$ .

f (0) = 0 + 0 + 34

$f (0) = 0 + 0 + 34$

f (0) = 0 + 0 + 34

$f (0) = 0 + 0 + 34$

Step 2.1.2.2

Simplify by adding numbers.

Tap for more steps...

Step 2.1.2.2.1

Add

0

$0$ and

0

$0$ .

f (0) = 0 + 34

$f (0) = 0 + 34$

Step 2.1.2.2.2

Add

0

$0$ and

34

$34$ .

f (0) = 34

$f (0) = 34$

f (0) = 34

$f (0) = 34$

Step 2.1.2.3

The final answer is

34

$34$ .

34

$34$

34

$34$

34

$34$

Step 2.2

Since

34 = 34

$34 = 34$ , the point is on the graph.

The point is on the graph

Step 3

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

m

$m$

=

$=$ The derivative of

f (x) = x^{2} + 3 x + 34

$f (x) = x^{2} + 3 x + 34$

Step 4

Consider the limit definition of the derivative.

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

Step 5

Find the components of the definition.

Tap for more steps...

Step 5.1

Evaluate the function at

$x = x + h$ .

Tap for more steps...

Step 5.1.1

Replace the variable

$x$ with

$x + h$ in the expression.

$f (x + h) = {(x + h)}^{2} + 3 (x + h) + 34$

Step 5.1.2

Simplify the result.

Tap for more steps...

Step 5.1.2.1

Simplify each term.

Tap for more steps...

Step 5.1.2.1.1

Rewrite

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

$(x + h) (x + h)$ .

$f (x + h) = (x + h) (x + h) + 3 (x + h) + 34$

Step 5.1.2.1.2

Expand

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

Tap for more steps...

Step 5.1.2.1.2.1

Apply the distributive property.

$f (x + h) = x (x + h) + h (x + h) + 3 (x + h) + 34$

Step 5.1.2.1.2.2

Apply the distributive property.

$f (x + h) = x \cdot x + x h + h (x + h) + 3 (x + h) + 34$

Step 5.1.2.1.2.3

Apply the distributive property.

$f (x + h) = x \cdot x + x h + h x + h \cdot h + 3 (x + h) + 34$

Step 5.1.2.1.3

Simplify and combine like terms.

Tap for more steps...

Step 5.1.2.1.3.1

Simplify each term.

Tap for more steps...

Step 5.1.2.1.3.1.1

Multiply

$x$ by

$x$ .

$f (x + h) = x^{2} + x h + h x + h \cdot h + 3 (x + h) + 34$

Step 5.1.2.1.3.1.2

Multiply

$h$ by

$h$ .

$f (x + h) = x^{2} + x h + h x + h^{2} + 3 (x + h) + 34$

Step 5.1.2.1.3.2

Add

$x h$ and

$h x$ .

Tap for more steps...

Step 5.1.2.1.3.2.1

Reorder

$x$ and

$h$ .

$f (x + h) = x^{2} + h x + h x + h^{2} + 3 (x + h) + 34$

Step 5.1.2.1.3.2.2

Add

$h x$ and

$h x$ .

$f (x + h) = x^{2} + 2 h x + h^{2} + 3 (x + h) + 34$

Step 5.1.2.1.4

Apply the distributive property.

$f (x + h) = x^{2} + 2 h x + h^{2} + 3 x + 3 h + 34$

Step 5.1.2.2

The final answer is

$x^{2} + 2 h x + h^{2} + 3 x + 3 h + 34$ .

$x^{2} + 2 h x + h^{2} + 3 x + 3 h + 34$

Step 5.2

Reorder.

Tap for more steps...

Step 5.2.1

Move

$3 x$ .

$x^{2} + 2 h x + h^{2} + 3 h + 3 x + 34$

Step 5.2.2

Move

$x^{2}$ .

$2 h x + h^{2} + x^{2} + 3 h + 3 x + 34$

Step 5.2.3

Reorder

$2 h x$ and

$h^{2}$ .

$h^{2} + 2 h x + x^{2} + 3 h + 3 x + 34$

Step 5.3

Find the components of the definition.

$f (x + h) = h^{2} + 2 h x + x^{2} + 3 h + 3 x + 34$

$f (x) = x^{2} + 3 x + 34$

$f (x + h) = h^{2} + 2 h x + x^{2} + 3 h + 3 x + 34$

$f (x) = x^{2} + 3 x + 34$

Step 6

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{h^{2} + 2 h x + x^{2} + 3 h + 3 x + 34 - (x^{2} + 3 x + 34)}{h}$

Step 7

Simplify.

Tap for more steps...

Step 7.1

Simplify the numerator.

Tap for more steps...

Step 7.1.1

Apply the distributive property.

$f' (x) = lim_{h \to 0} \frac{h^{2} + 2 h x + x^{2} + 3 h + 3 x + 34 - x^{2} - (3 x) - 1 \cdot 34}{h}$

Step 7.1.2

Simplify.

Tap for more steps...

Step 7.1.2.1

Multiply

$3$ by

$- 1$ .

$f' (x) = lim_{h \to 0} \frac{h^{2} + 2 h x + x^{2} + 3 h + 3 x + 34 - x^{2} - 3 x - 1 \cdot 34}{h}$

Step 7.1.2.2

Multiply

$- 1$ by

$34$ .

$f' (x) = lim_{h \to 0} \frac{h^{2} + 2 h x + x^{2} + 3 h + 3 x + 34 - x^{2} - 3 x - 34}{h}$

Step 7.1.3

Subtract

$x^{2}$ from

$x^{2}$ .

$f' (x) = lim_{h \to 0} \frac{h^{2} + 2 h x + 3 h + 3 x + 34 + 0 - 3 x - 34}{h}$

Step 7.1.4

Add

$h^{2}$ and

$0$ .

$f' (x) = lim_{h \to 0} \frac{h^{2} + 2 h x + 3 h + 3 x + 34 - 3 x - 34}{h}$

Step 7.1.5

Subtract

$3 x$ from

$3 x$ .

$f' (x) = lim_{h \to 0} \frac{h^{2} + 2 h x + 3 h + 0 + 34 - 34}{h}$

Step 7.1.6

Add

$h^{2}$ and

$0$ .

$f' (x) = lim_{h \to 0} \frac{h^{2} + 2 h x + 3 h + 34 - 34}{h}$

Step 7.1.7

Subtract

$34$ from

$34$ .

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

Step 7.1.8

Add

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

$0$ .

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

Step 7.1.9

Factor

$h$ out of

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

Tap for more steps...

Step 7.1.9.1

Factor

$h$ out of

$h^{2}$ .

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

Step 7.1.9.2

Factor

$h$ out of

$2 h x$ .

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

Step 7.1.9.3

Factor

$h$ out of

$3 h$ .

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

Step 7.1.9.4

Factor

$h$ out of

$h (h) + h (2 x)$ .

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

Step 7.1.9.5

Factor

$h$ out of

$h (h + 2 x) + h \cdot 3$ .

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

Step 7.2

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 7.2.1

Cancel the common factor of

$h$ .

Tap for more steps...

Step 7.2.1.1

Cancel the common factor.

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

Step 7.2.1.2

Divide

$h + 2 x + 3$ by

$1$ .

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

Step 7.2.2

Reorder

$h$ and

$2 x$ .

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

Step 8

Evaluate the limit.

Tap for more steps...

Step 8.1

Split the limit using the Sum of Limits Rule on the limit as

$h$ approaches

$0$ .

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

Step 8.2

Evaluate the limit of

$2 x$ which is constant as

$h$ approaches

$0$ .

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

Step 8.3

Evaluate the limit of

$3$ which is constant as

$h$ approaches

$0$ .

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

Step 9

Evaluate the limit of

$h$ by plugging in

$0$ for

$h$ .

$2 x + 0 + 3$

Step 10

Add

$2 x$ and

$0$ .

$2 x + 3$

Step 11

Simplify

$2 (0) + 3$ .

Tap for more steps...

Step 11.1

Multiply

$2$ by

$0$ .

$m = 0 + 3$

Step 11.2

Add

$0$ and

$3$ .

$m = 3$

Step 12

The slope is

$m = 3$ and the point is

$(0, 34)$ .

$m = 3, (0, 34)$

Step 13

Find the value of

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

Tap for more steps...

Step 13.1

Use the formula for the equation of a line to find

$b$ .

$y = m x + b$

Step 13.2

Substitute the value of

$m$ into the equation.

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

Step 13.3

Substitute the value of

$x$ into the equation.

$y = (3) \cdot (0) + b$

Step 13.4

Substitute the value of

$y$ into the equation.

$34 = (3) \cdot (0) + b$

Step 13.5

Find the value of

$b$ .

Tap for more steps...

Step 13.5.1

Rewrite the equation as

$(3) \cdot (0) + b = 34$ .

$(3) \cdot (0) + b = 34$

Step 13.5.2

Simplify

$(3) \cdot (0) + b$ .

Tap for more steps...

Step 13.5.2.1

Multiply

$3$ by

$0$ .

$0 + b = 34$

Step 13.5.2.2

Add

$0$ and

$b$ .

$b = 34$

Step 14

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.

$y = 3 x + 34$

Step 15

Enter YOUR Problem