Ps_final

PROBLEM SET 1

RATE OF CHANGE 1. Find the rate of which the volume of a right circular cylinder of constant altitude 10 feet changes with respect to its diameter when the radius is 5 feet. Ans: 50π ft3/ft 2. A spherical snowball increasing at rate if the surface area of snowball is 4.0 m2. How is the volume increase with respect to its surface area?

Ans:

√

3. Sand is pouring from a spout and is forming a cone whose altitude is always equal to the radius of the base. Find the rate of change of the volume with respect to the altitude when the latter is 15ft. Ans: 25π ft3/ft 4.

PROBLEM SET 2

RECTILINEAR MOTION: VELOCITY AND ACCELERATION 1. A ball is thrown vertically upward with a velocity of 48 ft/sec at the edge of a cliff 432 feet above the ground. What is the acceleration? Ans: - 32 ft/s2 2. A particle moves along a straight line according to the formula, ( ) position of the particle when acceleration is zero. Ans: 35 feet

in feet find the

3. A particle moves along a path with its position meters. (a) When does the velocity equal to 30 meters per second? (b) When is the particle at rest? Ans: t = 18.0s and t = 3.0 s 4. The vertical position of a ball is given by ( ) will reach? Ans: 194 meters

in meter. What is the maximum height the ball

5. Two particles are moving according to equations, and in meters, respectively. Find the respective positions of particles when their velocities are equal. Ans: 7.75 m and 14.0 m 6. The formula ( ) gives the height in meters of an object after it is thrown vertically upward from a point 15 meters above the ground at a velocity of 49.0 m/sec. How high above the ground will the object reach? Ans: 137. 5m 7. A basketball player throws a ball straight upwards at a velocity of 6.67 m/s from a height of 2.50 m above the floor. How many seconds will it be until the ball hits the floor? Ans: 1.7s 8. A ball is thrown vertically upward with a velocity of 40ft.per sec. from the top of a tower 200ft high. (a) find the maximum height attained by the ball (b) find the time it will take the ball to reach the ground. Ans: 224.8 feet and 5.14 s 9. A body moves according to the equation, is 3 m/s. Ans: 12.0 meters

. Determine the position of the body when the velocity

10. An object is thrown vertically upward with an equation of reached the object on the maximum height. Ans:

sec

determines the time required to

SLOPE OF A CURVE

PROBLEM SET 2

TANGENT AND NORMAL LINE at point, (

1. Find the equation of tangent line of the curve,

). Ans:

at point, (

2. Find the tangent and normal line equation of the curve

√

). Ans:

3. Find an equation of the tangent line to the curve, Ans:

that is parallel to the line,

4. Find an equation of the normal line to the curve,

that is parallel to the line,

.

Ans: 5. Find an equation of the tangent line to the curve, Ans:

that is parallel to the line

6. Find an equation of each of the normal lines to the curve, Ans: 7. Find an equation of the normal line to the curve, 8. Find an equation of the tangent line to the curve 9. Find an equation of the normal line to the curve, Ans: 10. At what point of the curve,

(

√

that is parallel to the line

at the origin. Ans: at the point (1, 2). Ans: at the point, (2, 3).

) is the tangent line parallel to x – axis. Ans: (1, 0)(1/3, 4/3)

√

PROBLEM SET 2

ANGLE OF INTERSECTION BETWEEN TWO LINES

PROBLEM SET 4

CRITICAL POINT Directions: Use first and second derivative tests in finding the critical points of the following functions. 1.

( )

Ans: (0, 2) maximum, (3, -25) minimum

2.

( )

Ans: (0, 2) maximum, (

3.

( )

Ans: (0, 0) minimum, (2, 0) minimum, (1, ¼) maximum

4.

( )

Ans: (1/8, - ¼ ) minimum

5.

( )

6.

( )

7.

( )

8.

( )

9.

( )

10.

( )

) minimum

√

(

)

POINT OF INFLECTION 1. If

and its point of inflection is at (

2. Find the point of inflection of the curve

) what is the value of .

–

–

. –

3. Find the point of inflection of the curve

–

4. Find the point of inflection of the curve 5. Find the point of inflection of the curve

. .

–

–

6. Locate the point of inflection of the curve

–

7. Find the points of inflection of the curve,

.

. –

– . Ans: (1, 1) (-1, 1)

CONCAVITY INCREASING AND DECREASING FUNCTIONS MAXIMA – MINIMA APPLICATIONS 1. A piece of wire 36 cm long is cut to make an equilateral triangle and rectangle with length is twice its width. Find the length of the rectangle, so that the sum of the areas of a triangle and rectangle is at minimum. Ans: 5.57 cm 2. Divide the number 120 into two parts such that the product P of one part and the square of the other is a maximum. Ans: 80, 40

TIME RELATED RATES 1. A spherical snowball is melting in such a way that its surface area decreases at the rate of 1.0cm2/min. How fast is its radius shrinking when it is 3.0 cm? Ans:

cm/min

2.

L’ HOSPITAL RULE APPROXIMATION AND ERRORS CENTER OF CURVATURE 1. Find the center of curvature of the curve, 2.

at the point (

). Ans: (

)