Calculus Answers

Domain and renge f(x)=1/2x-p

At any point (x, y) on a curve, 𝑑^2𝑦/𝑑𝑥^2 = 1-x^2

, and an equation of the tangent line to the

curve at the point (1, 1) is y = 2 – x. Find an equation of the curve.

{Fs} 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).

The line 𝑦=𝑥−5 is tangent to the curve 𝑦=𝑥^2−9𝑥−𝑘

Find the value of 𝑘.

Find dy/dx of the function xy² + e^6x y³ = cos(x² + 2) at the point (1,1) by using implicit differentiation. 

The area of a square is increasing at the rate of 28cm² s^-1. Find the increasing rate of the length of a side,x when the area of the square is 49cm² .

A knuckleball thrown with a different grip than that of the problem above has left/right

position as it crosses the plate given by

f(W) =

(0.625/w2)(1-sin (2.72w+π/2))

Again, use graphical and tabular evidence to make a conjecture about the lim f(w) as w approaches 0 from the right.

show that the equation of the tangent to x²+xy+y=0 at the point(x₁,y₁) is

(2x₁+y₁)x+(x₁+1) y+y₁ =0

A box with a square base and an open top is to have volume 62.5 m^3. Neglect the thickness of the material used to make the box and find the dimension that will minimize the amount of material used.