Calculus Answers

1. Bounded by the curve, the x-axis and given ordinates y=2x+1 from x=0 to x=4.

2. Bounded by the x-axis and the curve y=6x+x²-x³.

3. Bounded by y=5x and y=x³-4x.

4. Bounded by x²=y and x²-8y+4=0.

5. In the first quadrant bounded by x=0, y=6, x=5 and y²=5x.

1. Write the equation of the curve for which y'''=-96x, if the curve is tangent to the line 20x+y+1=0 at (1, -3) and passes thru (0, 1)?

2. A particle moves such that its acceleration is defined by the equation a=3t²-2. When t=0, s=5 and when t=1, v=-20. What is s when t=2?

3. A sandbag is dropped from a balloon rising at 9.6 m/s at a height of 64 m. Find the highest point reached by the sandbag and the time of flight. Note, use g=-10 m/s².

Solve the following recurrence relation using Recurrence Tree Method.

𝑇(𝑛) = ∫ 1 if n+1

∫T(n/2) + n if n >1

Show all the steps.

Determine the surface area of the solid obtained by rotating:

1) x² + y² = 16 from x = 2 to x = 4

2) y² = 12x from x = 0 to x = 3

3) y = x³ from x = 0 to x = 1

3. Let f be the function defined by

f (x) =( e^x/2)/x.

(a) Is there a y–intercept? (Explain)

(b) Determine the horizontal and vertical asymptotes (if any).

(c) Use the sign pattern for f^'(x) to determine

(i) the interval(s) over which f rises and where it falls;

(ii) the local extrema.

(d) Use the sign pattern for f

00 (x) to determine.

(i) where the graph of f is concave up and where it is concave down;

(ii) the inflection point(s) (if any). [20]

4. A rectangular sheet of metal of perimeter 36cm and dimensions x and y is to be rolled into a

cylinder. (Volume of cylinder is V = πr²h.) What values of x and y give the largest volume?

(The thickness of the metal can be neglected.)

How fast is the surface area of a spherical balloon increasing when the radius is 10 cm and

the volume is increasing at 15 cm³/sec? [7]

2. Let f be the function defined by

f (x) = xe^-x:

(a) Determine the y–intercept.

(b) Determine the horizontal and vertical asymptotes.

(c) Use the sign pattern for f^' (x) to determine

(i) the interval(s) over which f rises and where it falls;

(ii) the local extrema.

(d) Use the sign pattern for f^'' (x) to determine

(i) where the graph of f is concave up and where it is concave down;

(ii) the inflection point(s) (if any).

Find the volume of the solid revolution generated by rotating the curve y = x² bounded by x = 3 and y = 0; about the y-axis

Determine the area of the region bounded by:

1.) y = x², y = 2x + 3

2.) x² = y - 1, x = y - 3

3.) y = x², y = 2 - x²

Find the area using definite integration

1.) y = 3x² from x = 1 to x = 2

2.) xy = -1 from x = 1 to x = 2

3.) y = 4 - x² from x = -3 to x = 3

Determine the average value of f(x,y) = e^x+y over the region R=[0,2]X[0,2].