Are we related?
The radius of a circle is decreasing at a rate of 2.6 feet per second. This is the rate of change of the circumference at the instant the radius is 8 feet long.
What is the circumference is increasing at a rate of 5.2pi ft/sec?
For the function f(x)=x^(2/5) over [-3,1], these are absolute extrema and where they occur.
What is 9^(1/5) at x=-3 for the absolute max and 0 at x=0 for the absolute min?
What is sec2x?
The particle is moving to the left and has a positive acceleration. It is speeding this.
What is down?
Given that s(t) = 1/3t3 - 7/2t2 + 10t + 5, this is the displacement from t=1 to t=4.
What is -1.5?
The top of a 25 foot ladder, leaning against a vertical wall, is slipping down the wall at a rate of 1 ft/min. This is how fast the bottom of the ladder is slipping along the ground when the bottom of the ladder is 7 ft. away from the wall.
What is 24/7 ft/min?
Each statement is not always true for these reasons:
a) If f'(5) = 0, then there is a maximum or minimum at x=5.
b)If x=2 is a critical point, then f'(2)=0
What is that x=5 is a critical point but maxes and mins do not occur at every critical point and f'(2)=DNE is also a possibility?
sin(pi/3)=this.
What is sqrt(3)/2?
What is y+1 = -1/6(x-pi/4) or y=-1/6x+(pi-24)/24?
Given that s(t) = 1/3t3 - 7/2t2 + 10t + 5, this is when the acceleration is equal to 0.
What is t=7/2?
A 6 ft. tall man is 10 ft. away from an 18 ft. light pole. The man walks towards the pole at a rate of 1 ft/sec. This is the rate at which the length of his shadow is changing.
What is 1/2 ft/sec?
If f is a continuous, decreasing function on [0,10] with a critical point at (4,2) then this statement must be false:
A) f(10) is an absolute minimum of f on [0,10]
B) f(4) is neither a relative maximum nor minimum
C) f'(4) does not exist
D) f'(4) = 0
E) f'(4) < 0
Between (-pi/2,pi/2), the graph of y=cosx has horizontal tangents here.
What is x=0?
This is the derivative of the inverse function at x=2 if f(x)=x3 + 2x - 1 and f(1)=2.
What is 1/5?
This is the lim x--> 2 (x4-16)/(x-2).
What is 32?
Two cars start moving from the same point. One travels south at 60 mph and the other travels west at 25 mph. This is the rate that the distance between the cars are increasing two hours later.
What is 65 mph?
These are the critical numbers for f(x) if f'(x) = (4x3 + 13x2 + 3x)/(x-4).
What are x=0, -1/4, x=-3, x=4?
This is the equation of the tangent line to the curve y=4 + cot(x) - 2csc(x) at x=pi/2.
What is y-2 = -1(x - pi/2)?
This is the derivative of the inverse function if y=sin(x), -pi/2 <= x <= pi/2, and a=1/2.
What is 2/sqrt(3)?
This is the slope of sqrt(xy) = 1 at (2,1/2).
What is -1/4?
What is 40/81pi ft/min?
These are the absolute maximum and absolute minimum of the function y = t2/(t+3) over [-2,6].
What are abs max 4 at x=6 and x=2 and abs min 0 at x=0?
lim h-->0 (tan(pi/4 +h) - tan(pi/4))/h is equal to this.
What is 2?
If dy/dx = (3x2y - y2)/(2xy - x3), then this is the x-value where the tangent line to the curve is vertical. Note: xy2 - x3y = 6
What is (-24)^1/5?
Let f be a twice-differentiable function such that f(2) = 5 and f(5) =2. Let g be given by g(x) = f(f(x)).
This is why there must be a value c for 2<c<5 such that f'(c) = -1.
What is that the MVT guarantees that there is a value c, with 2<c<5, so that
f'(c) = (f(5) - f(2))/(5-2) = (2-5)/(5-2) = -1. Since it is differentiable on the entire function it must be continuous on the entire function.