Calculate the derivative of the following functions.

y = ⁴√(2x / (4x - 3))

Velocity fro​​​​m po​​sition ​The graph of $s=f\left(t\right)$ represents the position of an object moving along a line at time $t\ge 0$ . <IMAGE>

a. Assume the velocity of the object is 0 when $t=0$ . For what other values of t is the velocity of the object zero?

Velocity fro​​​​m po​​sition ​The graph of $s=f\left(t\right)$ represents the position of an object moving along a line at time $t\ge 0$ . <IMAGE>

c. Sketch a graph of the velocity function.

Calculate the derivative of the following functions.

y = cos⁴ θ + sin⁴ θ

Derivatives and tangent lines

a. For the following functions and values of a, find f′(a).

f(x) = 1/3x-1; a= 2

Fish length Assume the length L​​ (in centimeters) of a particular species of fish after t years is modeled by the following graph. <IMAGE>

b. What does the derivative tell you about how this species of fish grows?

Derivative calculations Evaluate the derivative of the following functions at the given point.

f(t) = 1/t+1; a=1

Use the definition of the derivative to determine d/dx (√ax+b), where a and b are constants.

Calculate the derivative of the following functions.

y = (2x⁶ - 3x³ + 3)²⁵

Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers. 

f(x) = (√x+1)(√x-1)

Calculate the derivative of the following functions.

y = (1 + 2 tan u)^4.5

Reproduce the graph of f and then plot a graph of f' on the same axes. <IMAGE>

Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers. 

f(w) = w³-w/w

The position (​in ​meters) of a marble, given an initial velocity and rolling up a long incline, is given by s = 100t / t+1, where t is measured in seconds and s=0 is the starting point.

b. Find the velocity function for the marble.

Calculate the derivative of the following functions.

g(x) = x / e^3x

{Use of Tech} A different interpretation of marginal cost Suppose a large company makes 25,000 gadgets per year in batches of x items at a time. After analyzing setup costs to produce each batch and taking into account storage costs, planners have determined that the total cost C(x) of producing 25,000 gadgets in batches of x items at a time is given by C(x) = 1,250,000+125,000,000 / x + 1.5x.

a. Determine the marginal cost and average cost functions. Graph and interpret these functions.

An angler hooks a trout and begins turning her circular reel at 1.5 rev/s. Assume the radius of the reel (and the fishing line on it) is 2 inches.

a. Let R equal the number of revolutions the angler has turned her reel and suppose L is the amount of line that she has reeled in. Find an equation for L as a function of R.

Calculate the derivative of the following functions.

y = sin(sin(e^x))

Where is the function continuous? Differentiable? Use the graph of f in the figure to do the following. <IMAGE>

a. Find the values of x in (0, 3) at which f is not continuous.

Where is the function continuous? Differentiable? Use the graph of f in the figure to do the following. <IMAGE>

b. Find the values of x in (0, 3) at which f is not differentiable.

Where is the function continuous? Differentiable? Use the graph of f in the figure to do the following. <IMAGE>

c. Sketch a graph of f'.

Robert Boyle (1627–1691) found that for a given quantity of gas at a constant temperature, the pressure P (in kPa) and volume V of the gas (in m³) are accurately approximated by the equation V=k/P, where k>0 is constant. Suppose the volume of an expanding gas is increasing at a rate of 0.15 m³/min when the volume V=0.5 m³ and the pressure is P=50 kPa. At what rate is pressure changing at this moment?

Evaluate and simplify y'.

x = cos (x−y)

A capacitor is a device in an electrical circuit that stores charge. In one particular circuit, the charge on the capacitor Q varies in time as shown in the figure. <IMAGE>

a. At what time is the rate of change of the charge Q' the greatest?

A capacitor is a device in an electrical circuit that stores charge. In one particular circuit, the charge on the capacitor Q varies in time as shown in the figure. <IMAGE>

b. Is Q′ positive or negative for t≥0?

An object oscillates along a vertical line, and its position in centimeters is given by y(t)=30(sint - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.​

At what times and positions is the velocity zero?

An object oscillates along a vertical line, and its position in centimeters is given by y(t) = 30(sin t - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.​

The acceleration of the oscillator is a(t) = v′(t). Find and graph the acceleration function.

Calculate the derivative of the following functions.

y = cos^7/4(4x³)

The energy (in joules) released by an earthquake of magnitude M is given by the equation E = 25,000 ⋅ 10^1.5M. (This equation can be solved for M to define the magnitude of a given earthquake; it is a refinement of the original Richter scale created by Charles Richter in 1935.)

Compute the energy released by earthquakes of magnitude 1, 2, 3, 4, and 5. Plot the points on a graph and join them with a smooth curve.

Power and energy are often used interchangeably, but they are quite different. Energy is what makes matter move or heat up. It is measured in units of joules or Calories, where 1 Cal=4184 J. One hour of walking consumes roughly 10⁶J, or 240 Cal. On the other hand, power is the rate at which energy is used, which is measured in watts, where 1 W = 1 J/s. Other useful units of power are kilowatts (1 kW=10³ W) and megawatts (1 MW=10⁶ W). If energy is used at a rate of 1 kW for one hour, the total amount of energy used is 1 kilowatt-hour (1 kWh = 3.6×10⁶ J) Suppose the cumulative energy used in a large building over a 24-hr period is given by E(t)=100t + 4t² − (t³ / 9) kWh where t = 0 corresponds to midnight.

The power is the rate of energy consumption; that is, P(t) = E′(t) Find the power over the interval 0 ≤ t ≤ 24.