Calculus Examples

$f (x) = \sin^{2} (x) - \cos (x)$ , and $0 \leq x \leq π$

Step 1

Find the critical points.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1.1

By the Sum Rule, the derivative of $\sin^{2} (x) - \cos (x)$ with respect to $x$ is $\frac{d}{d x} [\sin^{2} (x)] + \frac{d}{d x} [- \cos (x)]$ .

$\frac{d}{d x} [\sin^{2} (x)] + \frac{d}{d x} [- \cos (x)]$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [\sin^{2} (x)]$ .

Tap for more steps...

Step 1.1.1.2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \sin (x)$ .

Tap for more steps...

Step 1.1.1.2.1.1

To apply the Chain Rule, set $u$ as $\sin (x)$ .

$\frac{d}{d u} [u^{2}] \frac{d}{d x} [\sin (x)] + \frac{d}{d x} [- \cos (x)]$

Step 1.1.1.2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$2 u \frac{d}{d x} [\sin (x)] + \frac{d}{d x} [- \cos (x)]$

Step 1.1.1.2.1.3

Replace all occurrences of $u$ with $\sin (x)$ .

$2 \sin (x) \frac{d}{d x} [\sin (x)] + \frac{d}{d x} [- \cos (x)]$

Step 1.1.1.2.2

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$2 \sin (x) \cos (x) + \frac{d}{d x} [- \cos (x)]$

Step 1.1.1.3

Evaluate $\frac{d}{d x} [- \cos (x)]$ .

Tap for more steps...

Step 1.1.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \cos (x)$ with respect to $x$ is $- \frac{d}{d x} [\cos (x)]$ .

$2 \sin (x) \cos (x) - \frac{d}{d x} [\cos (x)]$

Step 1.1.1.3.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$2 \sin (x) \cos (x) - - \sin (x)$

Step 1.1.1.3.3

Multiply $- 1$ by $- 1$ .

$2 \sin (x) \cos (x) + 1 \sin (x)$

Step 1.1.1.3.4

Multiply $\sin (x)$ by $1$ .

$2 \sin (x) \cos (x) + \sin (x)$

Step 1.1.1.4

Simplify.

Tap for more steps...

Step 1.1.1.4.1

Reorder terms.

$2 \cos (x) \sin (x) + \sin (x)$

Step 1.1.1.4.2

Simplify each term.

Tap for more steps...

Step 1.1.1.4.2.1

Reorder $2 \cos (x)$ and $\sin (x)$ .

$\sin (x) (2 \cos (x)) + \sin (x)$

Step 1.1.1.4.2.2

Reorder $\sin (x)$ and $2$ .

$2 \cdot \sin (x) \cos (x) + \sin (x)$

Step 1.1.1.4.2.3

Apply the sine double-angle identity.

$f' (x) = \sin (2 x) + \sin (x)$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $\sin (2 x) + \sin (x)$ .

$\sin (2 x) + \sin (x)$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $\sin (2 x) + \sin (x) = 0$ .

Tap for more steps...

Step 1.2.1

Set the first derivative equal to $0$ .

$\sin (2 x) + \sin (x) = 0$

Step 1.2.2

Apply the sine double-angle identity.

$2 \sin (x) \cos (x) + \sin (x) = 0$

Step 1.2.3

Factor $\sin (x)$ out of $2 \sin (x) \cos (x) + \sin (x)$ .

Tap for more steps...

Step 1.2.3.1

Factor $\sin (x)$ out of $2 \sin (x) \cos (x)$ .

$\sin (x) (2 \cos (x)) + \sin (x) = 0$

Step 1.2.3.2

Raise $\sin (x)$ to the power of $1$ .

$\sin (x) (2 \cos (x)) + \sin (x) = 0$

Step 1.2.3.3

Factor $\sin (x)$ out of $\sin^{1} (x)$ .

$\sin (x) (2 \cos (x)) + \sin (x) \cdot 1 = 0$

Step 1.2.3.4

Factor $\sin (x)$ out of $\sin (x) (2 \cos (x)) + \sin (x) \cdot 1$ .

$\sin (x) (2 \cos (x) + 1) = 0$

Step 1.2.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\sin (x) = 0$

$2 \cos (x) + 1 = 0$

Step 1.2.5

Set $\sin (x)$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.5.1

Set $\sin (x)$ equal to $0$ .

$\sin (x) = 0$

Step 1.2.5.2

Solve $\sin (x) = 0$ for $x$ .

Tap for more steps...

Step 1.2.5.2.1

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$x = \arcsin (0)$

Step 1.2.5.2.2

Simplify the right side.

Tap for more steps...

Step 1.2.5.2.2.1

The exact value of $\arcsin (0)$ is $0$ .

$x = 0$

Step 1.2.5.2.3

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$x = π - 0$

Step 1.2.5.2.4

Subtract $0$ from $π$ .

$x = π$

Step 1.2.5.2.5

Find the period of $\sin (x)$ .

Tap for more steps...

Step 1.2.5.2.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 1.2.5.2.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 1.2.5.2.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 1.2.5.2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 1.2.5.2.6

The period of the $\sin (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 2 π n, π + 2 π n$ , for any integer $n$

Step 1.2.6

Set $2 \cos (x) + 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.6.1

Set $2 \cos (x) + 1$ equal to $0$ .

$2 \cos (x) + 1 = 0$

Step 1.2.6.2

Solve $2 \cos (x) + 1 = 0$ for $x$ .

Tap for more steps...

Step 1.2.6.2.1

Subtract $1$ from both sides of the equation.

$2 \cos (x) = - 1$

Step 1.2.6.2.2

Divide each term in $2 \cos (x) = - 1$ by $2$ and simplify.

Tap for more steps...

Step 1.2.6.2.2.1

Divide each term in $2 \cos (x) = - 1$ by $2$ .

$\frac{2 \cos (x)}{2} = \frac{- 1}{2}$

Step 1.2.6.2.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.6.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.6.2.2.2.1.1

Cancel the common factor.

$\frac{2 \cos (x)}{2} = \frac{- 1}{2}$

Step 1.2.6.2.2.2.1.2

Divide $\cos (x)$ by $1$ .

$\cos (x) = \frac{- 1}{2}$

Step 1.2.6.2.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.6.2.2.3.1

Move the negative in front of the fraction.

$\cos (x) = - \frac{1}{2}$

Step 1.2.6.2.3

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$x = \arccos (- \frac{1}{2})$

Step 1.2.6.2.4

Simplify the right side.

Tap for more steps...

Step 1.2.6.2.4.1

The exact value of $\arccos (- \frac{1}{2})$ is $\frac{2 π}{3}$ .

$x = \frac{2 π}{3}$

Step 1.2.6.2.5

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$x = 2 π - \frac{2 π}{3}$

Step 1.2.6.2.6

Simplify $2 π - \frac{2 π}{3}$ .

Tap for more steps...

Step 1.2.6.2.6.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = 2 π \cdot \frac{3}{3} - \frac{2 π}{3}$

Step 1.2.6.2.6.2

Combine fractions.

Tap for more steps...

Step 1.2.6.2.6.2.1

Combine $2 π$ and $\frac{3}{3}$ .

$x = \frac{2 π \cdot 3}{3} - \frac{2 π}{3}$

Step 1.2.6.2.6.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 3 - 2 π}{3}$

Step 1.2.6.2.6.3

Simplify the numerator.

Tap for more steps...

Step 1.2.6.2.6.3.1

Multiply $3$ by $2$ .

$x = \frac{6 π - 2 π}{3}$

Step 1.2.6.2.6.3.2

Subtract $2 π$ from $6 π$ .

$x = \frac{4 π}{3}$

Step 1.2.6.2.7

Find the period of $\cos (x)$ .

Tap for more steps...

Step 1.2.6.2.7.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 1.2.6.2.7.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 1.2.6.2.7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 1.2.6.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 1.2.6.2.8

The period of the $\cos (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{2 π}{3} + 2 π n, \frac{4 π}{3} + 2 π n$ , for any integer $n$

Step 1.2.7

The final solution is all the values that make $\sin (x) (2 \cos (x) + 1) = 0$ true.

$x = 2 π n, π + 2 π n, \frac{2 π}{3} + 2 π n, \frac{4 π}{3} + 2 π n$ , for any integer $n$

Step 1.2.8

Consolidate $2 π n$ and $π + 2 π n$ to $π n$ .

$x = π n, \frac{2 π}{3} + 2 π n, \frac{4 π}{3} + 2 π n$ , for any integer $n$

Step 1.3

Find the values where the derivative is undefined.

Tap for more steps...

Step 1.3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 1.4

Evaluate $\sin^{2} (x) - \cos (x)$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 1.4.1

Evaluate at $x = 0$ .

Tap for more steps...

Step 1.4.1.1

Substitute $0$ for $x$ .

$\sin^{2} (0) - \cos (0)$

Step 1.4.1.2

Simplify.

Tap for more steps...

Step 1.4.1.2.1

Simplify each term.

Tap for more steps...

Step 1.4.1.2.1.1

The exact value of $\sin (0)$ is $0$ .

$0^{2} - \cos (0)$

Step 1.4.1.2.1.2

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

$0 - \cos (0)$

Step 1.4.1.2.1.3

The exact value of $\cos (0)$ is $1$ .

$0 - 1 \cdot 1$

Step 1.4.1.2.1.4

Multiply $- 1$ by $1$ .

$0 - 1$

Step 1.4.1.2.2

Subtract $1$ from $0$ .

$- 1$

Step 1.4.2

Evaluate at $x = π$ .

Tap for more steps...

Step 1.4.2.1

Substitute $π$ for $x$ .

$\sin^{2} (π) - \cos (π)$

Step 1.4.2.2

Simplify.

Tap for more steps...

Step 1.4.2.2.1

Simplify each term.

Tap for more steps...

Step 1.4.2.2.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\sin^{2} (0) - \cos (π)$

Step 1.4.2.2.1.2

The exact value of $\sin (0)$ is $0$ .

$0^{2} - \cos (π)$

Step 1.4.2.2.1.3

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

$0 - \cos (π)$

Step 1.4.2.2.1.4

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$0 - - \cos (0)$

Step 1.4.2.2.1.5

The exact value of $\cos (0)$ is $1$ .

$0 - (- 1 \cdot 1)$

Step 1.4.2.2.1.6

Multiply $- (- 1 \cdot 1)$ .

Tap for more steps...

Step 1.4.2.2.1.6.1

Multiply $- 1$ by $1$ .

$0 - - 1$

Step 1.4.2.2.1.6.2

Multiply $- 1$ by $- 1$ .

$0 + 1$

Step 1.4.2.2.2

Add $0$ and $1$ .

$1$

Step 1.4.3

Evaluate at $x = \frac{2 π}{3}$ .

Tap for more steps...

Step 1.4.3.1

Substitute $\frac{2 π}{3}$ for $x$ .

$\sin^{2} (\frac{2 π}{3}) - \cos (\frac{2 π}{3})$

Step 1.4.3.2

Simplify.

Tap for more steps...

Step 1.4.3.2.1

Simplify each term.

Tap for more steps...

Step 1.4.3.2.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\sin^{2} (\frac{π}{3}) - \cos (\frac{2 π}{3})$

Step 1.4.3.2.1.2

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

${(\frac{\sqrt{3}}{2})}^{2} - \cos (\frac{2 π}{3})$

Step 1.4.3.2.1.3

Apply the product rule to $\frac{\sqrt{3}}{2}$ .

$\frac{{\sqrt{3}}^{2}}{2^{2}} - \cos (\frac{2 π}{3})$

Step 1.4.3.2.1.4

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 1.4.3.2.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}} - \cos (\frac{2 π}{3})$

Step 1.4.3.2.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3^{\frac{1}{2} \cdot 2}}{2^{2}} - \cos (\frac{2 π}{3})$

Step 1.4.3.2.1.4.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{3^{\frac{2}{2}}}{2^{2}} - \cos (\frac{2 π}{3})$

Step 1.4.3.2.1.4.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.4.3.2.1.4.4.1

Cancel the common factor.

$\frac{3^{\frac{2}{2}}}{2^{2}} - \cos (\frac{2 π}{3})$

Step 1.4.3.2.1.4.4.2

Rewrite the expression.

$\frac{3^{1}}{2^{2}} - \cos (\frac{2 π}{3})$

Step 1.4.3.2.1.4.5

Evaluate the exponent.

$\frac{3}{2^{2}} - \cos (\frac{2 π}{3})$

Step 1.4.3.2.1.5

Raise $2$ to the power of $2$ .

$\frac{3}{4} - \cos (\frac{2 π}{3})$

Step 1.4.3.2.1.6

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$\frac{3}{4} - - \cos (\frac{π}{3})$

Step 1.4.3.2.1.7

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$\frac{3}{4} - - \frac{1}{2}$

Step 1.4.3.2.1.8

Multiply $- - \frac{1}{2}$ .

Tap for more steps...

Step 1.4.3.2.1.8.1

Multiply $- 1$ by $- 1$ .

$\frac{3}{4} + 1 (\frac{1}{2})$

Step 1.4.3.2.1.8.2

Multiply $\frac{1}{2}$ by $1$ .

$\frac{3}{4} + \frac{1}{2}$

Step 1.4.3.2.2

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3}{4} + \frac{1}{2} \cdot \frac{2}{2}$

Step 1.4.3.2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 1.4.3.2.3.1

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

$\frac{3}{4} + \frac{2}{2 \cdot 2}$

Step 1.4.3.2.3.2

Multiply $2$ by $2$ .

$\frac{3}{4} + \frac{2}{4}$

Step 1.4.3.2.4

Combine the numerators over the common denominator.

$\frac{3 + 2}{4}$

Step 1.4.3.2.5

Add $3$ and $2$ .

$\frac{5}{4}$

Step 1.4.4

List all of the points.

$(0 + 2 π n, - 1), (π + 2 π n, 1), (\frac{2 π}{3} + 2 π n, \frac{5}{4})$ , for any integer $n$

Step 2

Exclude the points that are not on the interval.

$(0, - 1), (π, 1), (\frac{2 π}{3}, \frac{5}{4})$

Step 3

Evaluate at the included endpoints.

Tap for more steps...

Step 3.1

Evaluate at $x = 0$ .

Tap for more steps...

Step 3.1.1

Substitute $0$ for $x$ .

$\sin^{2} (0) - \cos (0)$

Step 3.1.2

Simplify.

Tap for more steps...

Step 3.1.2.1

Simplify each term.

Tap for more steps...

Step 3.1.2.1.1

The exact value of $\sin (0)$ is $0$ .

$0^{2} - \cos (0)$

Step 3.1.2.1.2

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

$0 - \cos (0)$

Step 3.1.2.1.3

The exact value of $\cos (0)$ is $1$ .

$0 - 1 \cdot 1$

Step 3.1.2.1.4

Multiply $- 1$ by $1$ .

$0 - 1$

Step 3.1.2.2

Subtract $1$ from $0$ .

$- 1$

Step 3.2

Evaluate at $x = π$ .

Tap for more steps...

Step 3.2.1

Substitute $π$ for $x$ .

$\sin^{2} (π) - \cos (π)$

Step 3.2.2

Simplify.

Tap for more steps...

Step 3.2.2.1

Simplify each term.

Tap for more steps...

Step 3.2.2.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\sin^{2} (0) - \cos (π)$

Step 3.2.2.1.2

The exact value of $\sin (0)$ is $0$ .

$0^{2} - \cos (π)$

Step 3.2.2.1.3

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

$0 - \cos (π)$

Step 3.2.2.1.4

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$0 - - \cos (0)$

Step 3.2.2.1.5

The exact value of $\cos (0)$ is $1$ .

$0 - (- 1 \cdot 1)$

Step 3.2.2.1.6

Multiply $- (- 1 \cdot 1)$ .

Tap for more steps...

Step 3.2.2.1.6.1

Multiply $- 1$ by $1$ .

$0 - - 1$

Step 3.2.2.1.6.2

Multiply $- 1$ by $- 1$ .

$0 + 1$

Step 3.2.2.2

Add $0$ and $1$ .

$1$

Step 3.3

List all of the points.

$(0, - 1), (π, 1)$

Step 4

Compare the $f (x)$ values found for each value of $x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest $f (x)$ value and the minimum will occur at the lowest $f (x)$ value.

Absolute Maximum: $(\frac{2 π}{3}, \frac{5}{4})$

Absolute Minimum: $(0, - 1)$

Step 5