Calculus Examples

$f (x) = 8.6 - 2.6 \cos (\frac{π}{153} t)$

Step 1

Find the first derivative of the function.

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Step 1.1

Combine $\frac{π}{153}$ and $t$ .

$\frac{d}{d x} [8.6 - 2.6 \cos (\frac{π t}{153})]$

Step 1.2

By the Sum Rule, the derivative of $8.6 - 2.6 \cos (\frac{π t}{153})$ with respect to $x$ is $\frac{d}{d x} [8.6] + \frac{d}{d x} [- 2.6 \cos (\frac{π t}{153})]$ .

$\frac{d}{d x} [8.6] + \frac{d}{d x} [- 2.6 \cos (\frac{π t}{153})]$

Step 1.3

Since $8.6$ is constant with respect to $x$ , the derivative of $8.6$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 2.6 \cos (\frac{π t}{153})]$

Step 1.4

Since $- 2.6 \cos (\frac{π t}{153})$ is constant with respect to $x$ , the derivative of $- 2.6 \cos (\frac{π t}{153})$ with respect to $x$ is $0$ .

$0 + 0$

Step 1.5

Add $0$ and $0$ .

$0$

Step 2

Since $0$ is constant with respect to $x$ , the derivative of $0$ with respect to $x$ is $0$ .

$f'' (x) = 0$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$0 = 0$

Step 4

Since $0 = 0$ , the equation will always be true.

Always true

Step 5

Evaluate the second derivative at $x =$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$0$

Step 6

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

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Step 6.1

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, \infty)$

Step 6.2

Substitute any number, such as $0$ , from the interval $(- \infty, \infty)$ in the first derivative $0$ to check if the result is negative or positive.

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Step 6.2.1

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

$f' (0) = 0$

Step 6.2.2

The final answer is $0$ .

$0$

Step 6.3

No local maxima or minima found for $f (x) = 8.6 - 2.6 \cos (\frac{π}{153} t)$ .

No local maxima or minima

Step 7