Calculus Answers

{F} The equation for a displacement 𝑠(𝑚), at a time 𝑡(𝑠) by an object starting at a displacement of 𝑠0 (𝑚), with an initial velocity 𝑢(𝑚𝑠 −1 ) and uniform acceleration 𝑎(𝑚𝑠 −2 ) is: 𝑠 = 𝑠0 + 𝑢𝑡 + 1 2 𝑎𝑡 2 A projectile is launched from a cliff with 𝑠0 = 30 𝑚, 𝑢 = 55 𝑚𝑠 −1 and 𝑎 = −10 𝑚𝑠 −2 . The tasks are to: a) Plot a graph of distance (𝑠) vs time (𝑡) for the first 10s of motion. b) Determine the gradient of the graph at 𝑡 = 2𝑠 and 𝑡 = 6𝑠. c) Differentiate the equation to find the functions for: i) Velocity (𝑣 = 𝑑𝑠 𝑑𝑡) ii) Acceleration (𝑎 = 𝑑𝑣 𝑑𝑡 = 𝑑 2 𝑠 𝑑𝑡2 ) d) Use your results from part c to calculate the velocity at 𝑡 = 2𝑠 and 𝑡 = 6𝑠. e) Compare your results for part b) and part d). f) Find the turning point of the equation for the displacement 𝑠 and using the second derivative verify whether it is a maximum, minimum or point of inflection. g) Compare your results from f) with the graph you produced in a).

{F} Determine the length of the parametric curve given by the following parametric equations. x=3sin(t) y=3cos(t) 0<t<2π

1. a metal washer 1 inch. in diameter is pierced by a 1/2 - inch hole. what is the area of one of the washer?
2. the vertical end of a trough has the following dimensions width at top 4.4ft., width at bottom 3.2ft., depth 3.5ft. find the area of the end of the trough.
3. the surface of hemispherical dome whose diameter is 36ft. is to be covered with gold leaf which costa 15 cents per square inch. what must be paid for the golf leaf?
4. a cubic foot of ivory weighs 114Ib. find the weight of 1000 ivory billiard balls 21 in. in diameter?

*SHOW SOLUTION*

1. the inside of a wash basin is in the shape of segment of a sphere; the distance across the top is 18in. and its greatest depth is 7in. find how many pints of water it will hold, reckoning 7.48gal to the cubic foot.

graphing p(x)=x⁶+2x⁵-2x³-x²

Write concepts and Application of Derivative and integral calculus.

5 A Ferris Wheel in Las Vegas, Nevada, opened in March 2014. The 550

ft tall wheel has a diameter of 5290 ft A ride on its one of its 28 passenger cars last 30 minutes the time it takes the wheel to complete one full rotation Riders board the passenger cars at the bottom of the wheel Assume that once the wheel is in motion it maintains a constant speed for the 30-minutes ride and is rotating in a counter clockwise direction. If you were on this ride how high would you be above the ground after 20 minutes?

"\\intop" 1/x²-4x+3 dx=?

"\\intop" ²_-1 1/ 3√x2 dx=?

"\\intop"₂⁴ e^x cosxdx=?