Calculus Answers

Find the unit tangent vector to any point on the curve
r=(t^2+1)I+(4t-3)j+(2t^2-6t)k

A door of a lecture hall is in a parabolic shape. The door is 56 inches across at the bottom of the door and parallel to the floor and 32 inches high. Sketch and find the equation describing the shape of the door. If you are 22 inches tall, how far must you stand from the edge of the door to keep from hitting your head

Find the slopes of the curve y x 1 0
2
− + = at the points )1 ,2( − and )1,2( .

If a chord of the parabola y^2 = 4ax is a normal at one of its ends, show that its mid-point lies on the curve 2(x−2a) = (y^2 /a) +( 8a^3/ y^2) . Prove that the shortest length of such a chord is 6a√3

Derive an equation x(t) for the instantaneous position of the car as a function of time. Identify the value x when t=0s.use the given data for the time to reach 400m to calculate a constant of interaction c
Formula:
V(t) =A(1-e^-t/tmax)
Numbers =
t(400m) (s) =10.5
Tmax=7.1s

Solve the following partial fractions
(i) 3x-1/2x(x^2 +2)

(ii) 2x+1/(x-1)(x^2 +1)

(iii) x^3 +x+1/x(x^2 -x+1)

(iv) 3x+2/(x-2)(x^2 +3x+1)

integrate x^2+x+5/(x^2+4)(x+1)dx

P(-1,-4) lies on the curve y=x^2+5x.

If Q is the point (x,x^2+ 5x), find the slope of the secant line PQ or x= -0.5

Suppose a particle P is moving in the plane so that its coordinates are given by P(x,y), where x = 4cos2t, y = 7sin2t. (i) By finding a,b ∈ R such that x2 a2 + y2 b2 = 1, show that P is travelling on an elliptical path. [10 marks] (ii) Let L(t) be the distance from P to the origin. Obtain an expression for L(t).[8 marks] (iii) How fast is the distance between P and the origin changing when t = π/8?[7 marks]

You throw a ball vertically upward from a height of 2 meters with an initial velocity of 10 meters per second. Use the idea that acceleration due to gravity (neglecting air resistance) is -32 feet/sec^2, which is the same as -9.8 meters/sec^2.

a. What is the maximum height the ball will reach? Round your answer jot the nearest meter, if necessary.

b. How many seconds will go by before the ball hits the ground after being thrown? Round your answer to the nearest tenth of a second, if necessary.