Horizontal Tangent line; Tangent of given slope a. Find the derivative f(x) and solve for values...

 

Transcribed Image Text:

Horizontal Tangent line; Tangent of given slope a. Find the derivative f(x) and solve for values where f(x) = 0 to identify the location of any horizontal tangent line. f(x) = 3x4 - 8x3 - 12x? + 5 { f(x) = 3x^4 - 8x^3 - 12x^2 +5 b. Find the derivative g'(x) and solve for all values where g'(x) = 1 to identify the location of tangent lines with slope m = 1. = 2 cos X g(x) = 2 cos x QUESTION 8 Free Fall Example (Constant acceleration from gravity) The position of an object in free fall (motion in which only force governing the motion is due to gravity) is given by r(t) as follows r(t) = - 16 t2 + 64 t + 80 { r(t) =- 16 t^2 + 64 t + 80 } Note, this position function corresponds to an object that begins at an initial height of 80 ft and given an initial upward velocity of 64 ft/sec. The only force acting on the object is the downward force of gravity. a) Find the velocity v(t) = r'(t). b) Observe where velocity is positive (motion in upward direction) and velocity is negative (motion in downward direction) When does the object reach its maximum height? What is the time when the object has its largest positive veocity (maximum of v(t))? its largest negative velocity (minimum of v(t))? c) Compute v(2), v(3), v(4), and v(5) Confirm that this is consistent with the action f gravity, which exerts a force that increase velocities by 32 ft/sec for each passing second Then show this coincides with the acceleration function a(t) = v'(t) = r"(t). {acceleration due gravity is 32 feet per second per second, meaning that for each second that passes, the velocity will increase by a magnitude of 32 feet per second) The direction of the force is toward the earth (represented by the (-) for the downward direction in the frame of reference given in the problem.)} QUESTION 9 Trigonometric Derivative Find the derivative of each of the following: a) f(x) = 3 sin x - 2 cos x { f(x) = 3 sin x - 2 cos x } b) g(x) = 8 tan x { g(x) = 8 tan x Horizontal Tangent line; Tangent of given slope a. Find the derivative f(x) and solve for values where f(x) = 0 to identify the location of any horizontal tangent line. f(x) = 3x4 - 8x3 - 12x? + 5 { f(x) = 3x^4 - 8x^3 - 12x^2 +5 b. Find the derivative g'(x) and solve for all values where g'(x) = 1 to identify the location of tangent lines with slope m = 1. = 2 cos X g(x) = 2 cos x QUESTION 8 Free Fall Example (Constant acceleration from gravity) The position of an object in free fall (motion in which only force governing the motion is due to gravity) is given by r(t) as follows r(t) = - 16 t2 + 64 t + 80 { r(t) =- 16 t^2 + 64 t + 80 } Note, this position function corresponds to an object that begins at an initial height of 80 ft and given an initial upward velocity of 64 ft/sec. The only force acting on the object is the downward force of gravity. a) Find the velocity v(t) = r'(t). b) Observe where velocity is positive (motion in upward direction) and velocity is negative (motion in downward direction) When does the object reach its maximum height? What is the time when the object has its largest positive veocity (maximum of v(t))? its largest negative velocity (minimum of v(t))? c) Compute v(2), v(3), v(4), and v(5) Confirm that this is consistent with the action f gravity, which exerts a force that increase velocities by 32 ft/sec for each passing second Then show this coincides with the acceleration function a(t) = v'(t) = r"(t). {acceleration due gravity is 32 feet per second per second, meaning that for each second that passes, the velocity will increase by a magnitude of 32 feet per second) The direction of the force is toward the earth (represented by the (-) for the downward direction in the frame of reference given in the problem.)} QUESTION 9 Trigonometric Derivative Find the derivative of each of the following: a) f(x) = 3 sin x - 2 cos x { f(x) = 3 sin x - 2 cos x } b) g(x) = 8 tan x { g(x) = 8 tan x