Question

Let h be the function defined by H(x)= integral pi/4 to x (sin^2(t))dt. Which of the following is an equation for the line tangent to the graph of h at the point where x= pi/4.

The function is given by H(x)= integral 1.1 to x (2+ 2ln( ln(t) ) -
( ln(t) )^2)dt for (1.1 < or = x < or = 7). On what
intervals, if any, is h increasing?

What is a left Riemann sum approximation of integral 2 to 8 (cos(x^2)dx with (n) sub-intervals of equal length?

Answer #1

Evaluate the integral.
pi/2
3
sin2(t) cos(t)
i + 5 sin(t)
cos4(t) j + 4
sin(t) cos(t)
k dt
0

a). Find dy/dx for the following integral.
y=Integral from 0 to cosine(x) dt/√1+ t^2 ,
0<x<pi
b). Find dy/dx for tthe following integral
y=Integral from 0 to sine^-1 (x) cosine t dt

Estimate the area under graph f(x) = sin(x) from x = 0 to x =
pi/2. Using 4 sub-intervals and the left endpoints. Sketch the
graph and the rectangles.

1. Integral sin(x+1)sin(2x)dx
2.Integral xe^x dx
3. integral ln sqrt(x) dx
4. integral sqrt(x) lnx dx

Let w(x,y,z) = x^2+y^2+z^2 where x=sin(8t), y=cos(8t) , z=
e^t
Calculate dw/dt by first finding dx/dt, dy/dt, and dz/dt and using
the chain rule
dx/dt =
dy/dt=
dz/dt=
now using the chain rule calculate
dw/dt 0=

1. If f(x) = ∫10/x t^3 dt then: f′(x)= ? and f′(6)= ?
2. If f(x)=∫x^2/1 t^3dt t then f′(x)= ?
3. If f(x)=∫x3/−4 sqrt(t^2+2)dt then f′(x)= ?
4. Use part I of the Fundamental Theorem of Calculus to find the
derivative of h(x)=∫sin(x)/−2 (cos(t^3)+t)dt. what is h′(x)= ?
5. Find the derivative of the following function:
F(x)=∫1/sqrt(x) s^2/ (1+ 5s^4) ds using the appropriate form of the
Fundamental Theorem of Calculus.
F′(x)= ?
6. Find the definitive integral: ∫8/5...

find g'(x)
g(x)= integral (-3/4 + t + cos(Pi/4 (t^2) + t)))
0<x<3

6.) Let ~r(t) =< 3 cos t, -2 sin t > for 0 < t < pi.
a) Sketch the curve. Make sure to pay attention to the parameter
domain, and indicate the orientation of the curve on your graph. b)
Compute vector tangent to the curve for t = pi/4, and sketch this
vector on the graph.

Using matlab.
Consider the function s(t) defined for t in [0,4) by
{ -sin(pi*t/2)+1 , for t in [0,1)
s(t) = { -(t-2)^5 , for t in [1,3)
{ sin(pi*t/2)-1 , for t in [3,4)
(i) Generate a column vector s consisting of 512 uniform
samples of this function over the interval [0,4). (This
is best done by concatenating three vectors.)

Find the slope of the line tangent to the function sin(2x)+ x at
x = pi/4

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