Calculus : Acceleration, Average Value and Values of Derivatives

8) The acceleration at time t, of a particle moving along the x axis is given by a(t)=20t^3+6. At time t=0 the velocity of the particle is 0 and the position of the particle is 7. What is the position of the particle at time t?

12) The function f is given by f(x)=3x^2+1. What is the average value os f over the closed interval [1,3]?

14) Selected values of function f and g and their derivatives, f' and g' are given in the table If h(x)=f(g(x)), what is h'(30)?

8. Suppose the position is s(t), velocity is v(t). We know s'(t)=v(t) and v'(t)=a(t)=20t^3+6. Thus v(t)=5t^4+6t+C. When t=0, v(0)=0. So C=0 and v(t)=5t^4+6t. ...

Solution Summary

Problems involving acceleration, average value and values of derivatives are solved.

Apply differentiation rules to find the derivatives of the functions.
Express the derivative dy/dx in terms of x without first rewriting y as a function of x.
1. y=u^5 and u=1/3x-2
Identify a function u of x and an integer such that f(x)=u^n. Then compute f '(x).
1. f(x)= ½+5x^3
2. f(x)=(x^2 - 4x

Find the exact length of the curve
x = 3*cos(t) - cos(3*t), y = 3*sin(t) - sin(3*t)
The curve is plotted in the attachment. How would I integrate the derivatives to get the length of the curve?
I think I have to integrate this in several steps, accounting first for the parts below the x-axis, and then for the curve above

Need help with calculus homework problems.
In Exercises 41-48, find dy/dx
44. 5x 4/5 + 10y 6/5 = 15
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54. a. By differentiating x 2 - y2 =1 implicitly, show that dy/dx = x/y
b. Then show that d2y/dx2 = - 1/y3.
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1.) (d/dx)(xe^(lnx^2))=?
2.) If x=e^(2t) and y=sin(2t), then (dy/dx)=?
3.) If y=xy+x^2+1, then when x=-1, (dy/dx) is ?
4.) A particle moves along the x-axis so that its acceleration at any time is a(t)=2t-7. If the initial velocity of the particle is 6, at what time t during the interval 0≤t≤4 is the particl

Please see the attached file for correctly formatted equations.
(3) Find the equation of the line tangent to g(x) = -6x + 64sqrt(x) at x = 16.
(4) Find the average rate of change of h(x) = -3x^4 + 14x^2 + 22x from x = -3 to x = 5.

An explanation with work would be greatly appreciated. Thank you.
Math Practice
1. For F (x,y) = e^(xcos(y)) find dF/dx, dF/dy, d^2F/dx^2, d^2F/dxdy
2. Normalize the Maxwell-Boltzmann distribution. More specifically, solve for N below:
(See attachment for full equations.)

Use part I of the Fundamental Theorem of Calculus to find the derivatives of the following functions; answers must use correct variable.
a. f(x)=the integral as pi goes to x of (1+cos[t])dt; f'(x)=___
b. f(u)=the integral as -1 goes to u of [1/(x+4x^2)]dx; f'(u)=___

This solution shows how to solve for various calculus problems, including differentiation of functions using the product rule, the quotient rule, and the chain rule, as well as how to calculate integrals.

Find the point(s) on the graph where the tangent line is horizontal...
50. Extend the product rule for differentiation to the following case involving the product of three differentiable functions...
Find derivative...
(Please see attached.)