Calculus Answers

Q.4) If an object starts out at rest and falls a distance of s feet in t seconds, then physical laws imply that s = 16t². Find ∆s/∆t as t changes from t0 to t0 + ∆t.

Use the result to find ∆s/∆t as t changes from:

(a) 3 to 3.5, (b) 3 to 3.2, and (c) 3 to 3.1

find the derivative using chain rule.

1. y= (x³ + 3)⁵

2. y=(-3x⁵ + 1)³

3. y=(-5x³ - 3)³

4. y=(5x² + 3)⁴

5. f(x)=⁴√-3x⁴-2

6. f(x)= √-2x² + 1

Suppose that a particle moves along a line with position function s(t) =2t^2 +3t+1 where s is in meters and t is in seconds.

a. What is its initial position?

b. Where is it located after t = 2 seconds?

c. At what time is the particle at position s = 6?

For what values of 𝑐 does the curve 𝑓(𝑥) = 2𝑥3 + 𝑐𝑥2 + 2𝑥 have maximum and minimum

points?

Show that the minimum and maximum points of every curve in the family of polynomials 

𝑓(𝑥) = 2𝑥3 + 𝑐𝑥2 + 2𝑥 lie on the curve 𝑦 = 𝑥 − 𝑥3. 

An automobile traveling at the rate of 20 m/s is approaching an intersection. When the automobile is 100 meters from the intersection, a truck traveling at the rate of 40 m/s crosses the intersection. The automobile and the truck are on perpendicular roads. How fast is the distance between the truck and the automobile changing two seconds after the truck leaves the intersection?

find 'f(t) by definition if f(t)=4t^2 + t also find tangent at t=2

The slope of the tangent line to the curve y = f(x) at (x,y) is y¹ = 10x²+5x. If (1,16) is a point on the curve

Use Lagrange Multipliers to find the maximum and minimum values of f(x,y)=xy

 subject to the constraint 4x^2+8y^2=16

.

Find the equation of the curve having y’ = 2x – 5 that passes through (5,4).