Derivatives 1
Find f'(x) for f(x)=3x2
f'(x)=6x
Say you differentiate a function f(x), then evaluate f ′ (x) it at a point x = a. If you get f ′ (a) = 0, what can you say about the tangent line of the graph of f(x) at that point?
The tangent line to the graph is horizontal at x = a.
What is the domain of f(x)=(x-5)/(x2-4)?
D: (-inf,-2)U(-2,2)U(2,inf)
Use a linear approximation to estimate √ 5.
f(x) = √ x
f'(x) = 1 /(2 √ x ), f'(4) = 1/4,
f(x) ≈ L(x) = f(4) +f'(4)(x − 4) = 2 + (1/4)(x − 4), L(5) = 9/4 ≈ √ 5
If a function f(x) = 3 at x = 2, does this mean that lim x→2 f(x) = 3? Why or why not?
No. There may be a discontinuity.
Find f'(x) for f(x)=e2x
f'(x)=2e2x
Use the definition of the derivative to find f ′ (x) where f(x) =x2 (verify this by using another derivative rule that you know)
f ′ (x) = 2x
Determine the critical points for f(x)=8x3+81x2−42x−8
x=−7, x=1/4
Find two positive numbers x and y such that x + 2y = 50 and (x + 1)(y + 2) is a maximum.
x = 53/2, y = 47/4
Find the limit at x = 4 (or state that it does not exist) of the following function: f(x) =x2 + 12x − 1
lim x→4 f(x) = 63
Find f'(x) for f(x)=arctan(2x)
f\(x)=2/(1+x2)
What is the 802nd derivative of sin(x)?
-sin(x)
Find the absolute maximum of f(x)=8x3+81x2−42x−8 on [−4,2].
944 at x=−4
Use L’Hopital’s Rule to find the following limit: lim x→∞ ln(3x)/x2
lim x→∞ f(x)=0
f(x) = [√ (x2+1)+5]/ (x−1). Find the limit as x → 1
Limit does not exist (negative infinity from the left and positive infinity from the right).
Find f'(x) for f(x)=ln(x)x3
f'(x)=3x2lnx+x2
Find the antiderivative of f(x)=x3+sinx
(x4/4)-cosx
Determine all the number(s) c which satisfy the conclusion of the Mean Value Theorem for f(x)=4x3−8x2+7x−2 on [2,5]
c=(2+√79)/3
A thin sheet of ice is in the form of a circle. If the ice is melting in such a way that the area of the sheet is decreasing at a rate of 0.5 m2/sec, at what rate is the radius decreasing when the area of the sheet is 12 m2 ?
r' = −0.040717
Evaluate the limit at x = 9 (or state that it does not exist) of the following function: f(x) = {? x − 4 if x <9,
x2 − 5x − 31 if x ≥ 9
lim x→9 f(x) = 5
Find f'(x) for f(x)=(2x−e8x)sin(2x)
f'(x)=sin(2x)ln(2x−e8x)
Find dy/dx for e2x+3y=x2-ln(xy3)
dy/dx=(2x-x-1-2e2x+3y)/(3e2x+3y+3y-1)
Determine the inflection points of f(x)=x4+12x3+6x2−36x+2
x=−3−2√2 & x=−3+2√2
We have a piece of cardboard that is 50cm by 20cm and we are going to cut out the corners and fold up the sides to form a box. Determine the height of the box that will give a maximum volume.
h = 4.4018
Find the limit as x → ∞ of the function f(x) = (x−4)/(2x2−11x+12)
limx→∞ f(x) = 0