Question

Let f(x) = x*(2-x) if x>=0, or x*(x+2) if x<0

i) graph the function from x=-3 to x=+3. If you like WolframAlpha, use Piecewise[{{x*(2-x),x=>0},{x*(x+2),x<0}}]

If you like Desmos, use f(x)= {x>=0:x*(2-x), x<0:x*(x+2)}

(for some reason, when you paste that it, it forgets the first curly-brace { so you’ll need to add it in by hand) Or, you can use this, but it makes it less clear how to take the derivative:

f(x) = -sign(x)*x*(x - 2*sign(x) )

ii) Find and graph its derivative, by hand. You might need to do some algebra before taking the derivative. Notice that you should take the derivative of the formula for when x<0 separately from the derivative for when x>0, but then graph it all on one graph.

iii) Find and graph its 2nd derivative

Answer #1

Let f(x, y) = −x 3 + y 2 . Show that (0, 0) is a saddle point.
Note that you cannot use the second derivative test for this
function. Hint: Find the curve of intersection of the graph of f
with the xz-plane.

Problem 43.)
a.) Use Excel to graph the function f(x) = x3 -6x2 + 9x -6 for
-2 ≤ x ≤ 5. (OK to draw the graph neatly on your homework, or cut
and paste from Excel.)
b.) Does it look the graph has both a relative maximum point
and relative minimum point? Estimate them from the graph.
c.) Find the points where the derivative f ’(x) = 0 and
compare to your answers from (b.)

1. Let f (x, y) =
xy((x^2-y^2)/(x^2+y^2)) if, (x, y) 6= (0, 0),
0, if (x, y) = (0, 0) (it's written as a piecewise function)
(a) Compute ∂f/∂y (0, 0) and ∂f/∂x (0, 0).
(b) Compute ∂f/∂y (x, 0) for all x, and ∂f/∂x (0, y) for all
y
(c) Use part (a) and (b) to compute ∂^2f/∂y∂x (0, 0) and ∂^2f/
∂x∂y (0, 0), then verify that:
∂^2f/∂y∂x (0, 0) does not equal ∂^f/ ∂x∂y (0, 0)

Let f ( x ) = cot x for 0 < x < π .
(a) State the range of f and graph it on the interval
given above.
(b) State the domain and range of g ( x ) = cot − 1 x . Graph
g ( x ) .
(c) State whether the graph increases or decreases on its
domain.
(d) Find each of the following limits based on the graph of g (
x...

1. Consider the function f(x) whose second derivative is
f″(x)=10x+2sin(x). If f(0)=2 and f′(0)=3, what is f(x)?

Consider the function f(x) whose second derivative is
f''(x)=2x+5sin(x). If f(0)=3 and f'(0)=2, what is f(5)?

1)Consider the function f(x)f(x) whose second derivative is
f″(x)=9x+8sin(x). If f(0)=4 and f′(0)=2, what is f(5)?
2) Consider the function f(x)=(7/x^3)−(2/x^5).
Let F(x) be the antiderivative of f(x) with F(1)=0.
3)Given that the graph of f(x) passes through the point (5,4)
and that the slope of its tangent line at (x,f(x)) is 3x+3, what
is f(2)?
Then F(2) equals

given the conditions for a function (f(x))
f '(x) > 0 on (0,3)
• f '(x) < 0 on (3,∞)
• f ''(x) > 0 on (0,2)
• f ''(x) < 0 on (−∞,0) ∪ (2,∞)
how would the graph f(x) look like?

Identify a specific function f that has the following
characteristics. Then draw it's graph.
f(0)=3
f'(x)=-2 for x E (-infinity, infinity).

Consider the function f(x) whose second derivative is
f′′(x)=6x+10sin(x). If f(0)=2 and f′(0)=4, what is f(x)?

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