Question

1. Consider the function

y= -2x^{3}+6x^{2}-2x+125

a. write the difference quotient of the function

b. demonstrate how to convert the difference quotient you found in part a into a derivative

Answer #1

For
the function y = -4x^3 + 1/2x^4 + 150 :
a. write down the formula of the function you are
analysing
b. find the intervals of increase/decrease
c. find the local max/min values, the intervals of concave
up/down and the points of inflection,
d.) sketch the graph of the function, ensuring that your graph
displays the feautures found above

consider the 2 variable function f(x,y) = 4 - x^2 - y^2 - 2x -
2y + xy
a.) find the x,y location of all critical points of f(x,y)
b.) classify each of the critical points found in part a.)

Consider the function f(x, y) = sin(2x − 2y) (a) Solve and find
the gradient of the function.
(b) Find the directional derivative of the function at the point
P(π/2,π/6) in the direction of the vector
v = <sqrt(3), −1>
(c) Compute the unit vector in the direction of the steepest
ascent at A (π/2,π/2)

1)Consider the curve y = x + 1/x − 1 .
(a) Find y' .
(b) Use your answer to part (a) to find the points on the curve
y = x + 1/x − 1 where the tangent line is parallel to the line y =
− 1/2 x + 5
2) (a) Consider lim h→0 tan^2 (π/3 + h) − 3/h This limit
represents the derivative, f'(a), of some function f at some number
a. State such an...

) Consider the function f(x,y)=−2x^2−y^2.
Find the the directional derivative of ff at the point (1,−3)(1,−3)
in the direction given by the angle θ=π/2.
Find the unit vector which describes the direction in which ff
is increasing most rapidly at (1,−3).

1) Write and answer a question that asks to solve an equation
that involves an logarithmic function. That is, your unknown must
be in a logarithmic function so that you must use inverse
operations (including exponentiation) to solve for it. Your
solution must include an exact answer, not a decimal
approximation.
2) Write and answer a question that asks you to find the
derivative of the sum or difference of at least three functions.
Your question and solution must demonstrate...

For the function y = x 3 − 2x 2 − 1, use the first and second
derivative tests to
(a) determine the intervals of increase and decrease.
(b) determine the local (relative) maxima and minima.
(c) determine the intervals of concavity.
(d) determine the points of inflection.
(e) sketch the graph with the above information indicated on the
graph.

Consider the function f (x) = x/(2x+1)*2 .
(i) Find the domain of this function. (Start by figuring out any
forbidden values!)
(ii) Use (i) to write the equation of the vertical asymptote for
this function.
(iii) Find the limits as x goes to positive and negative
infinity,
(iv) Find the derivative of this function.
(v) Find the coordinates at point A(..,…), where the
x-coordinate is 1. Use exact fractions, never a decimal
estimate.
(vi) Find the equation of the...

1) Write and answer a question that asks to find the derivative
of an implicitly defined equation. Your question and solution must
demonstrate include each of the following:
a. Use implicit differentiation to find ??/?? ,
b. Make the implicit equation from part a. explicit (use algebra
to define ? explicitly in terms of ?) and find ??/ ?? , and
c. Show that the solutions from part a. and part b. are the
same.
2) Write and solve a...

Consider the function f(x) =
x^2/x-1 with f ' (x) =
x^2-2x/ (x - 1)^2 and f ''
(x) = 2 / (x - 1)^3 are given. Use these to
answer the following questions.
(a) [5 marks] Find all critical points and determine the
intervals where f(x) is increasing and where it
is decreasing, use the First Derivative Test to fifind local
extreme value if any exists.
(b) Determine the intervals where f(x) is
concave upward and where it is...

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