Limits
Inf Limits/Continuity
Basic Derivatives
Product/Quotient Rule
Chain Rule
100
Consider the function shown below:
What is
lim_{x\to-1} f(x)?
Does not exist. DNE
100
What are the three types of discontinuity?
Removable, jump and infinite.
100
The limit below represents f'(a). Find some f and a .
\lim_{h\to0}\frac{cos(pi+h)+1}{h}
f(x) = cos(x),a=pi.
100
What is the formula for
[f(x)g(x)]'
f'(x)g(x)+g'(x)f(x)
100
What is the formula for
[f(g(x))]'
f'(g(x))g'(x)
200
Evaluate
\lim_{x\to -3}\frac{2x^2+5x-3}{x^2+5x+6}
7
200
Evaluate
\lim_{x\to1}\frac{2-x}{(x-1)^2}
-\infty
200
What is the derivative of f(x) = 3x^2-4x^{-1/2}+5cos(x)
f'(x) = 6x+2x^{-3/2}-5sin(x)
200
What is the formula for [f(x)/g(x)]'
\frac{g(x)f'(x)-f(x)g'(x)}{g(x)^2}
200
Evaluate
\cos(4x)
-4sin(4x)
300
Evaluate
\lim_{x\to 0}\frac{\sqrt{x+4}-2}{3x}
1/12
300
Evaluate
\lim_{x\to\infty}x-\sqrt{x}
\infty
300
The equation of motion of a particle is s = 2t^3-7t^2+4t+1
What is the acceleration after 1 second?
-2
300
Evaluate
\frac{d}{dx}[xcosx]
cos(x)-xsin(x)
300
Evaluate
d/dx[sin^2(3x^2+1)]
2\sin(3x^2+1)*(cos(3x^2+1))*(6x)
400
If
-x^2\leqx^2\cos(50pix)\sin(30pix)\leqx^2
then
\lim_{x\to0}\cos(50pix)\sin(30pix)=?
0
400
Evaluate
\lim_{x\to\infty}\frac{(2x+1)^2}{(x-1)^2(x^2+x)}
4
400
A particle has position function given by s = 5t^2-13sin(t)+2cos(t)
What is the instantaneous velocity of the particle at time t ?
v(t) = 10t-13cos(t)-2sin(t)
400
Evaluate
\frac{d}{dx}[x^2/sin(x)]
(2xsin(x)-x^2cos(x))/sin^2(x)
400
Evaluate
d/dx[xcos(x^2)]
x*(-sin(x^2)*2x)+cos(x^2)
500
Recall that
\lim_{x\to0}\frac{sin(x)}{x} = 1
Use this to evaluate
\lim_{x\to0}\frac{3\sin^2(x)\cos(x)}{x^2}
3
500
What are the horizontal and vertical asymptotes of
f(x)=1/x?
x = 0, y=0
500
Find d^8/dx^8[sin x]
sin(x)
500
Prove that using the product or quotient rule that
d/dx[sec x] = sec x tan x
d/dx[sec x] = d/dx[1/cos(x)] = (0 + sin(x))/(cos^2(x)) = sec(x)tan(x)
500
Find the tangent line to the curve y=sin(cos(x)) at
x=pi/2
y = -x+pi/2