Integrals/Derivatives
The derivative of y=1/2x4+3x2-x+e
What is y'=2x3+6x-1?
y=x2-4x+7 is continuous and differentiable for (-infinity, infinity)
What is the Mean Value Theorem?
All polynomials are continuous and differentiable over any domain.
The rate of change of radius with time
What is dr/dt?
d/dx[f(x)/g(x)]=g(x)f’(x)-f(x)g’(x)/[g(x)]2
What is the quotient rule?
The limit as y approaches 2 of (y2+5y+6)/(y+3)
What is (22+5(2)+6)/2+3=20/5=4?
y=2x3+x2 is increasing on intervals
What is (-infinity, infinity)
y'=6x2+2x, 2x(3x+1), x=1/3, + before and after 1/3
Find the minimum value for p(x)=-x2-24x on the interval [-6, -1].
What is P(-1)=23?
p'(x)=-2x-24=0, x=-12 (not in interval), P(-6)=108, P(-1)=23 (minimum)
The derivative of the volume of a sphere (V=4/3pir3)
What is 4/3pi(3r2dr/dt)?
f’(x)(g(x))g’(x)
What is the chain rule?
The limit as t approaches 3 of (t2-2t-3)/(t2-9)
What is 2/3?
Lim as t approaches 3 of (t-3)(t+1)/(t+3)(t-3)
Lim as t approaches 3 of (t+1)/(t+3)=4/6=2/3
What is (x2/2)-3ln|x|+c?
integral of ((x2/x)-(3/x))dx, integral of (x-(3/x))dx
Determine the 3 x-value candidates for absolute extrema on the interval [-3, 1] for the function g(x)=x2+4x.
What is f(-3), f(1), f(-2)?
g'(x)=2x+4, 2x+4=0, x=-2, endpoints
Air is leaking out of an inflated balloon in the shape of a sphere at a rate of 230pi cubic centimeters per minute. At the instant when the radius is 4 cm, which, if any, of the variables in the problems remain remain constant?
What is none?
The formula to find the derivative of an inverse
What is 1/f’(f-1(x))?
The limit as x approaches 3 from the left of x/(8-x)
What is 2.9/(8-2.9)…+/+…+♾
The derivative of y=ex+e-ex^2
What is y'=ex-2xex^2?
dy/dx=ex+0-2x(ex^2)
If g(2)=12 and g'(x) is greater than or equal to 1/2 on the interval (2,6), what is the smallest value g(6) can be?
What is g(6) is greater than or equal to 14?
(g(6)-g(2))/6-2 is greater than equal to 1/2, (g(6)-12)/4 is greater than or equal to 1/2, g(6)-12 is greater than or equal to 2, g(6) is greater than or equal to 14.
A rectangle’s base remains 0.5 cm while the height changes at a rate of 1.5 cm/min. At what rate is the area changing in cm2/min when the height is 1.5 cm?
What is 0.75 cm2/min?
A=b*h, A=0.5cmh, dA/dt=0.5cm(dh/dt), dA/dt=0.5cm(1.5cm/min)= 0.75 cm2/min.
The integral from initial time to final time of |v(t)|dt
What is distance?
The limit as x approaches 0 of sin/3x2-x
What is -1?
Lim as x approaches 0 of sinx/x(3x-1). Apply L’Hospitals Rule. Lim as x approaches 0 of cos/6x-1=cos(0)/6(0)-1=1/-1=-1
Integral of tan3xsec2xdx from 0 to pi/6
What is 1/36?
u=tanx, du= sec2x
Integral of u3xdx= 1/4(u)4 , 1/4(tanx)4 from 0 to pi/6, ((1/4)tan(pi/6))4-((1/4)tan(0))4=1/36
The function g(x) is continuous on the closed interval [8, 10] and differentiable on the open interval (8, 10). The value x=8.5 satisfies the conditions of the MVT on the interval [8, 10]. Find g(10) given g'(8.5)=-9 and g(8)=6.
What is g(10)=-12?
Avg ROC=Inst ROC, (g(10)-g(8))/10-8=-9, (g(10)-6)/2=-9, g(10)-6=-18, g(10)=-12.
A rectangle’s base remains 0.5 cm while it’s height changes at a rate of 1.5 cm/min. At what rate is the perimeter changing, in cm/min, when the height is 1.5 cm?
What is 3 cm/min?
P=2b+2h, P=2(0.5cm)+2h, dP/dt=0+2dh/dt, dP/dt=2(1.5cm/min)=3 cm/min.
The Accumulation Function
What is F(x)=f(a) + the integral from a to x of f’(t)dt?
Let g and h be the functions defined by g(x)=-x2-2x+3 & h(x)=1/2x2+x+13/2. If f is a function that satisfies g(x) is less than or equal to f(x) is less than or equal to h(x) for all x, what is the limit as x approaches -1 of f(x)?
What is the limit cannot be determined from the given information.
Lim as x approaches-1 of -x2-2x+3 is less than or equal to the Lim as x approaches -1 of f(x) is less than or equal to the Lim as x approaches -1 of 1/2x2+x+13/2.
4 is less than or equal to Lim as x approaches -1 of f(x) is less than or equal to 6.