Question

(a)(3p) Discuss the continuity of the following functions:

(i)(1.5p)

f(x)= (cosx if x<0, 2 if x=0, 1-x^4 if x>0)

at x=0

(ii)(1.5p)

f(x)= 5-2x if x<-3, x^2 + 2 if x>= -3

at x=−3

Answer #1

Consider the following functions
f(x) =x^2, g(x) = lnx, h(x) = cosx
For each of the following parts, you may use compositions,
products, and sums of thefunctions above, but no others. For
example, we can combine in the following waysh(g(x)) = cos(lnx), or
g(x)h(x) = lnxcosx, or g(x) +h(x) = lnx+ cosx
show how derivative rules apply to the function you came up
within order to produce the requested derivative.
1)A functionk(x) whose derivative is k′(x) = −tanx=
-(sinx/cosx)
2)...

13.1.7. Problem. Let f(x) = sinx and g(x) = cosx for 0 ≤ x ≤ π.
Find du(f,g) in the set of functions B([0, π]).
13.1.8. Problem. Let f(x) = 3x−3x3 and g(x) = 3x−3x2 for 0 ≤ x ≤ 2.
Find du(f,g) in the
set of
functions B([0, 2]).

For each of the following functions fi(x), (i) verify that they
are legitimate probability density functions (pdfs), and (ii) find
the corresponding cumulative distribution functions (cdfs) Fi(t),
for all t ? R.
f1(x) = |x|, ? 1 ? x ? 1
f2(x) = 4xe ?2x , x > 0
f3(x) = 3e?3x , x > 0
f4(x) = 1 2? ? 4 ? x 2, ? 2 ? x ? 2.

1- For the following functions, find all critical numbers
exactly.
f(x) = x − 2 sin x for −2π < x < 2π
f(x) = e^−x −e^−3x for x > 0
f(x) = x^5 − 2x^3

3. For each of the piecewise-defined functions f, (i) determine
whether f is 1-1; (ii) determine whether f is onto. Prove your
answers.
(a) f : R → R by f(x) = x^2 if x ≥ 0, 2x if x < 0.
(b) f : Z → Z by f(n) = n + 1 if n is even, 2n if n is odd.

1.
What is the derivative of f(x) = cosx/sinx
2. What is the derivative of f(x) = cos^-1 (3x)
3. What is the second derivative of tanx/secx
4. True or False: If f'(x) = 2^x, then a possible equation for
f is f(x) = 2^x +3
5. True or False: The equation x^2 + y^2 = 100 is an implicit
curve

What is the domain of the following functions?
(A) F(x)= x/sqrt (2x-4)
(B) F(x)= x-3/ sqrt(x^2-6x+9)

5. Part I
If f(x) = 2x 2 - 3x + 1, find f(3) - f(2).
A. 0
B. 7
C. 17
Part II
If G( x ) = 5 x - 2, find G-1(x).
A. -5x + 2
B. (x + 2)/5
C. (x/5) + 2
Please help me solve this problem and if you can please also
show or explain how you got that answer. Thank you! :)

Determine the intervals on which the following functions are
increasing and decreasing.
a) f(x) = 12 + x - x^2
b) g(x) = 2x^5 - (15x^4/4) + (5x^3/3)

1.
4(x+1)^2+5(x+1)-8 =
4(-x)^2+5(-x)-8=
4(x^3)^2+5(x^3)-8=
2.
f(x)= { x^2 if x<0. x+9 if x>=0}
f(-2) =
f(-1) =
f(0)=
f(1)=
f(2)=
3. f(x)= {x^2 +8x if x<= -1. X If -1< x<=1. -1 if
x>b1}
F(-3)=
F(-3/2)=
F(-1) =
F(40) =
4.
F(x)= 6x-7
f(2x)=
2f(x)=

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