Derivatives MC Calculator

The volume of a sphere is decreasing at a rate of 3 cm³/sec. At the instant when the radius of the sphere is decreasing at a rate of 0.25 cm/sec, what is the radius of the sphere? V=4/3πr³

• 0.141 cm

• 0.244 cm

• 0.250 cm

• 0.489 cm

• 0.977 cm

The figure above shows the graph of f ', the derivative of the function f. If f(0) = 0, which of the following could be the graph of f?

• 
• 
• 
• 
• 

The derivative of the function f is given by f '(x) = x³ - 4sin(x²) + 1. On the interval (-2.5, 2.5), at which of the following values of x does f have relative maximum?

• -1.970 and 0

• -1.467 and 1.075

• -0.475, 0.542, and 1.396

• -0.475 and 1.396 only

• 0.542

If f '(x) > 0 for all x and f ''(x) < 0 for all x, which of the following could be a table of values for f?

• 
• 
• 
• 
• 

A particle moves along the x-axis so that at time t ≥ 0 its position is given above. What is the velocity of the particle at the first instance the particle is at the origin?

• -1

• -0.624

• -0.318

• 0

• 0.065

Let f be the function with first derivative given by f '(x) = (3 - 2x - x²) sin (2x - 3). How many relative extrema does f have on the open interval -4 < x < 2?

• Two

• Three

• Four

• Five

• Six

The graph of a twice-differentiable function f is shown in the figure above. Which of the following is true?

• f '(-1) < f '(1) < f '(0)

• f '(-1) < f '(0) < f '(1)

• f '(0) < f '(-1) < f '(1)

• f '(1) < f '(-1) < f '(0)

• f '(1) < f '(0) < f '(-1)

A particle moves along a straight line so that at time t > 0 the position of the particle is given by s(t), the velocity is given by v(t), and the acceleration is given by a(t). Which of the following expressions gives the average velocity of the particle on the interval [2, 8]?

• 
• 
• 
• 
• 

The function f is differentiable and increasing for all real numbers x, and the graph of f has exactly one point of inflection. Of the following, which could be the graph of f ', the derivative of f?

• 
• 
• 
• 
• 

The table above gives selected values of a function f. The function is twice differentiable with f ''(x) > 0. Which of the following could be the value of f '(3)?

• 0.6

• 0.7

• 0.9

• 1.2

• 1.5