Limits
Find lim x-> infinity (7x5+1)/(5x7+1)
0
f(x) = sin(pi*x). What is the slope of the secant line to f(x) from x=0 to x= 3/2?
Setup: (sin(3pi/2)-sin(0))/(3/2-0) = -2/3
For the function f(x) = mx + b, what does the graph of the derivative look like? Be as specific as possible.
y = m, horizontal line
Find g'(x) if g(x) = 3x2 - e3
g'(x) = 6x
Describe three ways the graph of a function can be discontinuous.
Jump, infinite, or removeable discontinuity
Find lim t->5 (2x2-12x+10 )/(25-x2)
-4/5
Write the equation for the line tangent to g(x)=2/x3 where x=2.
y - 1/4 = -3/8 (x-2)
Graph a function where lim x->3 f(x) < f(3)
Answers vary
If y = 2xlnx, what is dy/dx?
dy/dx = 2+2lnx
If f(x) = 34x-1, find f'(0.5)
f'(x) = 34x-1(ln3)(4)
f'(0.5)=(3)(ln3)(4) = 12ln3
f(x) is piecewise defined:
f(x)= 2(x-1)3 - 8 if x<0
f(x)= -10ex if x>0
What is lim x->0 f(x)?
lim x->0 f(x) = -10
At what x-value does the graph of f(x) = 2xex have a horizontal tangent?
y' = 2xex + 2ex
0 = 2ex(x+1)=
x = -1
Graph a function where (lim x->infinity f(x)) = 1 and (lim x->-infinity f(x)) = -1
Need a graph where the horizontal asymptote on the left is y=-1 and the horizontal asymptote on the right is y=1.
If y= 5x2 - ex, what is y''(0)?
y'(x) = 10x - ex
y''(x) = 10 - ex
y''(0) = 9
If f(x) = sin(x+1)-g(x) and g(x)= 4x3 and h(x) = f(g(x)), find h'(x) (You don't need to simplify)
f(x)=sin(x+1)-4x3
f(g(x))=sin(4x3+1)-4(4x3)3
h'(x) = cos(4x3+1)(12x2)-4(3)(4x3)2(12x2)
Find lim x->4- g(x) where g(x) = (8-3x2)/(x2-16)
lim x-> 4- g(x) = infinity
What is the normal line to y=x3+x at x=2?
y-10 = -1/13(x-2)
How is it possible to have a graph continuous for all real numbers but not differentiable on the same intervals.
The graph can have a corner, cusp, or vertical tangent line.
If h(1)=5, h'(1)=2, g'(1)=9 and g(1)=-4, what is the derivative of 2h(x)g(x) evaluated at x=1?
Use the product rule: 2h'(1)g(1)+g'(1)2h(1) =2(2)(-4)+(9)(2)(5) =-16+90 =74
What is the y(5) if y=x6/30 in terms of x?
y(5)=24x
Find a value for "b" so that the piecewise function k(x) is continuous:
k(x) = (16-x2)/(20+5x), x does not equal -4
k(x) = b(x+2), x = -4
b=-4/5
What is the slope of the tangent line to
y = (3x-5)/(2x-3) at x=0?
1/9
Find and classify all x-coordinates on the piecewise functions that are either not continuous or not differentiable.
https://drive.google.com/file/d/1zpF00awfw5lxr75wPRNp7r6FvJMke48_/view?usp=sharing
x=0 infinite discontinuity and not differentiable
x=1 removeable discontinuity and not differentiable
Use limit definition of a derivative to find the derivative of f(x) = (x-3)0.5
f'(x) = lim h->0 ((x+h-3)0.5-(x-3)0.5)/h
f'(x) = lim h->0 ((x+h-3)-(x-3))/(h((x+h-3)0.5+(x-3)0.5)
f'(x) = lim h->0 (h/(h((x+h-3)0.5+(x-3)0.5)
f'(x) = lim h->0 (1/((x+h-3)0.5+(x-3)0.5)
f'(x) = 1/(2(x-3)0.5)
Given the position of a particle is s(t)=2x3-6x2+6x-5, find the time intervals where the particle is moving forward if t>0.
(0,1) u (1,infinity)