Review Skill 6.3 #1 Applications of Derivatives
Name: _______________________________
Answer on your notebook Review for 6.3 (page 1): Kinematics 1. For an object P, what do we mean (in terms of displacement, velocity and acceleration) when we say : a. P is at the origin b. P is at rest c. P’s velocity is increasing d. P is located to the right of the origin e. P is moving to the right/forward f.
P’s velocity is at a max, or min
g. P is moving to the left h. P is located to the left of the origin i.
P’s velocity is decreasing
j.
P is speeding up if…
k. P is slowing down if… 2. A ball is thrown vertically upwards into the air. The height, h metres, of the ball above the ground after t seconds is given by h = 2 + 20t – 5t2, t 0 (a) Find the initial height above the ground of the ball (that is, its height at the instant when it is released). (b) Show that the height of the ball after one second is 17 metres. (c) At a later time the ball is again at a height of 17 metres.
(i)
Write down an equation that t must satisfy when the ball is at a height of 17 metres.
(ii) Solve the equation algebraically.
(d)
(i)
(ii) Find the initial velocity of the ball (that is, its velocity at the instant when it is released).
(iii) Find when the ball reaches its maximum height.
Find d h . dt
(iv) Find the maximum height of the ball. 3. A particle P moves along the x axis with position given by x (t ) 1 2 cos t cm where t is time in seconds a. State the initial position, velocity and acceleration b. Describe the motion when t = 0.25 seconds c. Find the times when the particle reverses direction on 0 t 2 and find the position of the particle at these instants d. When is the particle’s speed increasing on 0 t 2
Review for 6.3 (page 2): Implicit Differentiation and Related Rates
Do you know the difference between implicitly and explicitly defined functions? What is it? Do you know that implicit differentiation is just an extension of the Chain Rule? Do you know how to differentiate implicitly and explicitly? Do you know all rules of derivatives? Do you know what the problem‐solving method for related rates problems is? Describe it.
Use implicit differentiation to find the following derivatives with respect to x:
1. Find the equation of the tangent line to x 2 y 2 9 at the point 2, 5 . 2. Find y ' if x 3 y 5 3 x 8 y 3 1 3. Find y ' if x 2 tan y y10 sec x 2 x
4. Find y ' if e 2 x 3 y x 2 ln xy 3 Use Related Rates to solve the following problems: 5. A 15 foot ladder is resting against the wall. The bottom is initially 10 feet away from the wall and is being pushed towards the wall at a rate of
1 ft/sec. How fast is the top of the ladder moving up the wall 12 seconds 4
after we start pushing? 6. A spot light is on the ground 20 ft away from a wall and a 6 ft tall person is walking towards the wall at a rate of 2.5 ft/sec. How fast is the height of the shadow changing when the person is 8 feet from the wall? Is the shadow increasing or decreasing in height at this time? 7. Two people on bikes are separated by 350 meters. Person A starts riding north at a rate of 5 m/sec and 7 minutes later Person B starts riding south at 3 m/sec. At what rate is the distance separating the two people changing 25 minutes after Person A starts riding? 8. A tank of water in the shape of a cone is leaking water at a constant rate of 2 ft 3 / hr . The base radius of the tank is 5 ft and the height of the tank is 14 ft. (a) At what rate is the depth of the water in the tank changing when the depth of the water is 6 ft? (b) At what rate is the radius of the top of the water in the tank changing when the depth of the water is 6 ft?
Review for 6.3 (page 3): Optimization and Properties of Curves
Do you know the Optimization problem solving method? Describe it. Can you show with calculus where a function is increasing/decreasing on an interval? Do you know what the three types of stationary points are, how to determine which type of point you have, and what the shape the curve has near the stationary point? Can you show with calculus where there is a point of inflection (stationary/ non‐stationary)? Can you show with calculus if a function is concave up/down? Do you know how to use first and/or second derivatives to determine if you are in the presence of a local max/min? Do you know what global maximum/minimums are? 1. If a function is increasing on an interval S, then f '( x ) ________ , and vice versa. 2. If a function is decreasing on an interval S, then f '( x ) ________ , and vice versa. 3. A function is monotone increasing/decreasing on the domain if: 4. A stationary point of a function is a point such that f '( x ) ___. 5. A sign diagram of f '( x ) with w/ f ( x ) having a local max at x a looks like this: 6. A sign diagram of f '( x ) with w/ f ( x ) having a local max at x a looks like this: 7. A sign diagram of f '( x ) with w/ f ( x ) having a horizontal point of inflection at x a looks like this:
or
8. If f ''( x ) 0 for all x S , the curve is _____________________ on the interval S 9. If f ''( x ) 0 for all x S , the curve is _____________________ on the interval S 10. If f ''( a ) 0 AND the sign of f ''( x ) changes on either side of x a then there is a ________________ at x a 11. To check for global max/min I need to check _________________________________ and compare them with the y‐values of the local max/min on that interval. 12. A curve has equation y = (x – 1)(x + 3)2. (a)
For this curve find (i)
the x‐intercepts;
(ii)
the coordinates of the maximum point;
(iii)
the coordinates of the point of inflexion.
(iv)
Sketch a graph of f, f’, and f”, showing the parts you found above.
13. Find the absolute maximum and minimum of f (t ) 4t 3 5t 2 8t 3 on 1,3 . 14. A box with a square bottom will be built to contain 40,000 cubic feet of grain. The sides of the box cost 10¢ per square foot to build, the roof costs $1 per square foot to build, and the bottom will cost $7 per square foot to build. What dimensions will minimize the building costs? 15. A printed page will have a total area of 96 square inches. The top and bottom margins will be 1 inch each, and the left and right margins will be 1.5 inches each. What overall dimensions for the page will maximize the area of the space inside the margins?