Derivatives
f (x) = 4x^3 − 3x^2 + 2x − π
f ' (x) = 12x^2 - 6x + 2
Find dy/dx of x^2 - 5xy + 3y^2 = 7
(2x-5y)/(5x-6y)
The rate of change of radius r of a circle is 4 cm/s. Find the rate of change of Area A when r=2cm.
What is 16π?
lim x-> 1 of (x^2 - 7x + 10)/(x^2 - 4)
4/-3
What is the Definition of a Derivative?
f '(x)=lim h->0 [f(x+h) - f(x)] / h
f (x) = (x^2)/3 - 3/(x^2)
f ' (x) = (2/3)x + (6/x^3)
Find dy/dx of sin(x/y) = 1/2
y/x
Find the average acceleration of a particle over the interval (0,50) given v(50)=80ft/sec and an initial velocity of 10 ft/sec. Include units in your answer.
What is 7/5 ft/sec2?
lim x->infinity of (5x^2+2)/sqrt(x^2+3)
DNE
What is L'Hospital's Rule? (formal notation!)
If the limx->a(f(x)/g(x)) is of indeterminate form 0/0 or inf/inf, then: limx->a(f(x)/g(x)) = limx->af'(x)/limx->ag'(x)
f (x) = √x - 1/√x
f ' (x) = 1/(2√x) + 1/(2x√x)
Find dy/dx of 4x^2 - 9y^2 = 17
(4x)/(9y)
The radius r of a sphere is increasing at a rate of 3 inches per minute. Find the rate of change of the volume when r = 9 inches, and r = 1 yard (leave both in terms of pi)
a) 972pi in^3/min
b) 15552pi in^3/min
The limit as x approaches 4 from the right of (4-x)/(x^2-16)
-1/8
What is the Intermediate Value Theorem?
If f(x) is continuous on the interval [a,b], f(a)=/=f(b), and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c)=k.
Find the derivative of f(x)=3x^2+7x-2 using the limit process
f ' (x) = 6x+7
Find dy/dx of ytan(x+y) = 4
y/x
A man six feet tall walks at a rate of 5 feet per second away from a light that is 15 feet above the ground. a) when he is 10 feet from the base of the light, at what rate is the tip of his shadow moving? b) when he is 10 feet from the base of the light, at what rate is the length of the shadow changing?
a) 25/3 ft/s
b) 10/3 ft/s
The limit as x approaches 0 of (cosx-1)/x
0
What are the three components of the Continuity Test?
1) f(c) is defined
2) lim x->c f(x) exists
3) lim x->c f(x) = f(c)
f(x)=sin^2(tan(2x))
f'(x)=2sin(2tan(2x))sec^2(2x))
Find the solution when dy/dx = cos(x) / y^2 , where y(π/2) = 0
y = (3 sin(x) - 3)^(1/3)
A ladder 25 feet long is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of 2 ft/s.
a) How fast is the top of the ladder moving down the wall when its base is 7 feet from the wall?
b) Find the rate at which the area of the triangle is changing when the base of the ladder is 7 feet from the wall.
a) -7/12 ft/s
b) 527/24 ft^2/s
limit as x approaches infinity of sin(2x)/x
0
What is the Mean Value Theorem?
If f is differentiable for all values of x in (a, b) and f is continuous at x=a and x=b, then there's at least one number x=c in (a, b) such that f'(c) = [f(b) - f(a)] / b-a