Limits
Evaluate lim_{x\to 1} (x^2-3x-2)
=-4
Evaluate
lim_{x\to -\infty} \frac{x^4-3x^3+x-1}{3x^5+2x^4-x^2+2}
=0
For a differentiable function f(x), state the limit definition of the derivative function f'(x)
f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}
For a function y=f(x), state the definition of the tangent line through the point (a,f(a))
The tangent line is the line through (a,f(a)), having slope f'(a)
This team won the 2023 super bowl
Kansas City Chiefs
Evaluate
lim_{x\to 2} \frac{x^2+x-6}{x^2-3x+2}
=5
Evaluate lim_{x\to \infty} \frac{2x-3}{-3|x|+7}
=-2/3
For y=f(x), state all four acceptable notations for the derivative
y', f'(x), \frac{dy}{dx}, \frac{d}{dx}(f(x))
Find the equation of the tangent line to f(x)=x^2+1 at x=1
y=2x
On the show "Friends", this character wrote the song "Smelly Cat"
Phoebe
Evaluate lim_{x\to -3} \frac{2x^2+3x-9}{x^2+5x+6}
=9
Find lim_{x\to \infty}(\frac{8}{2+3e^{-x}}+1)
=5
Evaluate the derivative of f(x)=\frac{1}{x^2}-\sqrt[x}-83
f'(x)=\frac{-2}{x^3}-\frac{1}{2\sqrt{x}}
Give an explicit example of a function or relation and a point where the tangent line is vertical (a graph is acceptable)
Once possible answer is the tangent line at x=1 for
x^2+y^2=1
You can't see him
John Cena
Evaluate lim_{x\to 7} \frac{\sqrt{x}-\sqrt{7}}{x-7}
=\frac{1}{2\sqrt{7}}
Evaluate lim_{x\to 3} \frac{-2}{(x-3)^8}
=-\infty
Find the derivative of g(x)= root(4)(x^3)-\frac{2}{root(3)(x)}
g'(x)=\frac{3}{4root(4)(x)}+\frac{2}{3root(3)(x^4)}
Find the equation of the tangent to f(x)=\sqrt{x} at x=4
y=\frac{1}{4}x+1
In "The Last of Us", he is responsible for bringing Ellie to the Fireflies
Joel
Give an explicit example of a function f(x) where lim_{x\to -2} f(x) does NOT exist
Many different possible answers, but one example is
f(x)=1/(x+2)
Evaluate lim_{x\to 2^-}\sqrt{4-x^2}
=0
Using the limit definition of a derivative, evaluate the derivative of f(x)=x^2
f'(x)=2x
Give an explicit example of a function and a point where the tangent line does NOT exist
One example is |x| at x=0
The first air-to-air kill for the F-22
Shooting down the Chinese balloon