Derivatives: Optimization Quiz

Derivatives are the mathematical tool used to calculate the rate of change of a function, such as position with respect to time. This makes them essential for understanding motion and change.

The derivative of ax, where a is a constant, is simply a. This follows the rule that the derivative of a constant multiplied by a variable is the constant itself.

To differentiate y = x^3 + 2x, apply the power rule. The derivative of x^3 is 3x^2 and the derivative of 2x is 2. Thus, the derivative is 3x^2 + 2, which is the correct choice.

To find the second derivative of f(x) = 2x - 5x^6, first compute the first derivative: f'(x) = 2 - 30x^5. Then, differentiate again to get f''(x) = -150x^4. Thus, the correct answer is f''(x) = -150x^4.

To find the derivative of x^4 cos(x), use the product rule: (u'v + uv'). Here, u = x^4 and v = cos(x). The derivative is 4x^3 cos(x) - x^4 sin(x). Simplifying gives x^4 cos(x) + 4x^3 sin(x), which matches the correct choice.

To find the maximum and minimum of 3 sin x + 4 cos x + 10, we rewrite it as R sin(x + φ) + 10, where R = √(3² + 4²) = 5. Thus, max = 15 and min = 5. Hence, the correct answer is 15 and 5.

To find the instantaneous velocity, we differentiate x(t) = -t² + 1. The derivative is v(t) = -2t. At t = 1, v(1) = -2(1) = -2. Thus, the instantaneous velocity at t = 1 is v = -2.

To find acceleration, differentiate the velocity function v = 2t + 3t^3. The derivative is a = dv/dt = 2 + 9t^2. At t = 2 s, a = 2 + 9(2^2) = 2 + 36 = 38 m/s². Thus, the correct answer is 38 m/s/s.

The picture contains three extrema: two local maxima and one local minimum. This is determined by analyzing the peaks and troughs in the graph, confirming that the correct answer is 3.

The graph has a minimum value because it reaches a lowest point, indicating that the function does not go below this value. Therefore, the correct answer is 'minimum'.