Limits
Secants and Tangents
Graphing
Derivatives
Miscellaneous
100
Find lim x-> infinity (7x^5+1)/(5x^7+1)
0
100
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
100
Sketch a graph of y=(x-1)^2 +3
https://www.desmos.com/calculator/aq6fws1xuh
100
is g(x) = |x|/x differentiable at x=0? Explain.
No, there is a jump discontinuity. You can't be differentiable if you're not continuous.
100
h(x) is continuous. Must h(x) be differentiable?
No, it could have corners, cusps, or vertical tangents.
200
Find lim t->5 (2x^2-12x+10 )/(25-x^2)
-4/5
200
Write the equation for the line tangent to g(x)=2/(x^3) where x=2.
y - 1/4 = -3/8 (x-2)
200
Graph a function where lim x->3 f(x) < f(3) < lim x->2- f(x) < lim x->2+ f(x)
Answers vary
200
If y = xlnx, what is dy/dx?
dy/dx = 1+lnx
200
Who are the two inventors of calculus?
Newton and Leibniz
300
f(x) is piecewise defined: f(x)= 2(x-1)^3 - 8 if x<0 f(x)= -10e^x if x>0 What is lim x->0 f(x)?
lim x->0 f(x) = -10
300
At what x-value does the graph of f(x) = 2xe^x have a horizontal tangent?
y' = 2xe^x + 2e^x = 0 2e^x(x+1)=0 x = -1
300
Graph a function where (lim x->infinity f(x)) = 1 + (lim x->-infinity f(x))
Need a graph where the horizontal asymptote on the left is one unit lower than the horizontal asymptote on the right.
300
If y= 5x^2 - e^x, what is y''(0)?
y'(x) = 10x - e^x y''(x) = 10 - e^x y''(0) = 9
300
If f(x) = sin(x+1)-g(x) and g(x)= 4x^3 what is f(g(x)) in terms of x? (You don't need to simplify)
f(x)=sin(x+1)-4x^3 f(g(x))=sin(4x^3+1)-4(4x^3)^3
400
Find lim x->4- g(x) (that's a limit from the left) where g(x) = (8-3x^2)/(x^2-16)
lim x-> 4- g(x) = negative / negative zero = infinity
400
What is the normal line to y=x^3+x at x=2?
y-10 = -1/13(x-2)
400
Sketch a graph of y = -sin(x-pi)+2 on [-pi, 2pi]
https://www.desmos.com/calculator/x12cqu0ykg
400
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
400
What is the Leibniz notation for the 8th derivative of g(k) evaluated at 7?
d^8g/dk^8|k=7
500
Find a value for b so that the piecewise function k(x) is continuous k(x) = (16-x^2)/(20+5x), x does not equal -4 b(x+2), x=-4
b=-4/5
500
What is the slope of the tangent line to y = (3x-5)/(2x-3) at x=0?
1/9
500
Sketch the piecewise function f(x) = ln(-x), x<0 1/(x-3) , x>0
https://www.desmos.com/calculator/ujxezxrqe4
500
lim h->0 (9e^(x+h)-6x-6h-9e^x+6x)/h represents the derivative of what function?
f(x) = 9e^x-6x
500
What is the domain of f(x) = 1 / (x ln(1-x^2))
(-1, 0) u (0, 1)
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