Applications of Derivatives
Math ⇒ Calculus
Applications of Derivatives starts at 11 and continues till grade 12.
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See sample questions for grade 12
A ball is thrown vertically upward with a velocity of 20 m/s. The height at time t seconds is given by h(t) = 20t - 5t². Find the time when the ball reaches its maximum height.
A box with a square base and open top must have a volume of 32,000 cm³. Find the dimensions that minimize the amount of material used.
A particle moves along a line so that its position at time t is given by s(t) = t³ - 6t² + 9t. Find the time when the particle is at rest.
A rectangle has a perimeter of 20 units. What is the maximum possible area of the rectangle?
A ball is thrown vertically upward with a velocity of 20 m/s. The height at time t seconds is given by h(t) = 20t - 5t². Find the time when the ball reaches its maximum height.
A box with a square base and open top must have a volume of 32,000 cm³. Find the dimensions that minimize the amount of material used.
A function f(x) has a local minimum at x = 1 and a local maximum at x = 3. Which of the following must be true? (1) f'(1) = 0 and f'(3) = 0 (2) f'(1) > 0 and f'(3) < 0 (3) f'(1) < 0 and f'(3) > 0 (4) f'(1) ≠ 0 and f'(3) ≠ 0
A particle moves along a line so that its position at time t is given by s(t) = t³ - 6t² + 9t. Find the time when the particle is at rest.
A function f(x) has a local minimum at x = 1 and a local maximum at x = 3. Which of the following must be true? (1) f'(1) = 0 and f'(3) = 0 (2) f'(1) > 0 and f'(3) < 0 (3) f'(1) < 0 and f'(3) > 0 (4) f'(1) ≠ 0 and f'(3) ≠ 0
A function has a local maximum at x = a if: (1) f'(a) = 0 and f''(a) > 0 (2) f'(a) = 0 and f''(a) < 0 (3) f'(a) = 0 and f''(a) = 0 (4) f'(a) ≠ 0
If f''(x) > 0 for all x in an interval, then the function f(x) is: (1) Concave up (2) Concave down (3) Linear (4) Constant
The rate of change of the volume of a sphere with respect to its radius r is: (1) 4πr² (2) 3πr² (3) 2πr (4) 4πr³
A function is said to be ______ at a point if its derivative does not exist at that point.
The derivative of a function at a point gives the ______ of the tangent to the curve at that point.
The point where the concavity of a function changes is called a(n) ______ point.
The process of finding the maximum or minimum value of a function is called ______.
If the derivative of a function is zero at a point, does it always mean the function has a maximum or minimum at that point?
True or False: The derivative of a position function with respect to time gives the velocity.
True or False: The first derivative test can be used to determine whether a critical point is a maximum or minimum.
True or False: The maximum or minimum value of a function on a closed interval always occurs at a critical point or at an endpoint.
