#1 Write an equation of the line tangent to the curve y=f(x) at the given point P on the curve. Express the answer in the form ax+by=c.
1)y=3x^2-4; P(1,-1)
2)y=2x-1/x; P(0.5,-1)

#2 Give the position function x=f(t) of a particle moving in a horizontal straight line. Find its location x when its velocity v is zero.
1)x=-16t^2=160t+25

#3 Give the height y(t) (in feet at time t seconds) of a ball thrown vertically upward. Find the maximum height that the ball attains.
1)y=-16t^2+128t+25

#4 Evaluate the Limits
1)lim as h goes to 0= 1/h(1/sqrt 9+h - 1/3)
2)lim x goes to 0= (sqrt 1+x - sqrt 1-x)/x

#5 Find a slope-predictor function for the given function f(x). Then write an equation for the line tangent to the curve y=f(x) at the point where x=2.
1)f(x)=x/x+1
2)f(x)=x^2+3/x
3)f(x)=x^2/x+1

Solution Preview

1.)In this case you need to calculate the derivative y'=f'(x) of each function. The derivative gives you the value of the slope of the line tangent to f(x) at point (x).
Once you have the derivative, you find the slope at the point you have been given. For instance in the first function you have to find f'(1) because 1 is the x coordinate of the point P you have been given. In the second case it would be f'(0.5).
Once you find the value of the slope you have all you need to calculate the equation of the line. You got a point P and a slope value.
The equation of a line is given by y=mx+b where m is the slope and x and y represent the coordinates of ...

Solution Summary

Topics included in this solution set are: write an equation of a tangent line, give the position function of a particle, give the height of a ball thrown upwards, evaluate limits, and find a slope-predictor function.

Please see the attached file for the fully formatted problems.
1. Use an iterated integral to find the area of a region...
2. Evaluate the double integral...
3. Use double integral to find the volume of a solid...
4. Verify moments of inertia...
5. Limit of double integral...
6. Surface area...
7. Triple integral...

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

Using the Fundamental Theorem of Calculus I need to find the solution of the following problems. Can you explain how?
Please see the attached file for the fully formatted problems.

We have the function f(x,y) = e^(-x^2 + y^2) :
a. To draw some level curves;
b. Calculate the gradient and determine the stationary points;
c. To write the equation of the tangent plane to z = f(x,y) in the point (0, 0, f(0, 0)), and in the point A (1,1,f(1,1));
d. To write the Taylor formula of f(x,y) stopped to the 2nd ord

** Please see the attached file for the complete problem description **
14. f(x) = (x^3 + 2x+ 1) (2+ (1)/(x^2)) Find derivative
34. Suppose f and g are functions that are differentiable at x = 1 and that:
f(1) = 2
f prime (1) = -1
g(1) = -2
g prime (1) = 3
Find the Value of h prime (1) for : h(x

I need help with these two problems. If you could please explain the solution, I would appreciate it.
1.Use cylindrical shells to compute the volume. The region bounded by y = x and y = x2 - 2, revolved about x = 3
2. A solid is formed by revolving the given region about the given line. Compute the volume exactly if possi

Please help me with the following calculusproblems:
1) Set up (do not integrate) an integral for the length of the curve y=tan-1x for x E [0,π).
2) Find the surface area obtained by rotating the curve x=2-y2 around the y axis.
3) Find the centroid of the region bounded by the curve x=2-y2 and the y axis.

12) What is limit as h approaches 0 of [cos(pi/2 + h) - cos(pi/2)] / [h]
Ans is -1. Explain.
14) The area of the region in the first quadrant between the graph of y=x times the
sqrt of (4-x^2) and the x axis is? Ans is 8/3. Explain.
15) If x^2 + y^3 = x^3y^2, then dy/dx = ? Explain.
Ans is [3x^2y^2 - 2x]

Let p1 = (4,0,4), p2 = (2,-1,8), and p3 = (1,2,3).
a) Show that the three points define a right triangle. Hint the difference between two vertices is a vector whose direction coincides with that of a triangle side, and a pair of such vectors must be orthogonal in order for the triangle to be a right triangle.
b) Specify