Calculus Answers

Find the slope and the equation of the tangent of the curve at the given point. Show your solution.

1) y = 2x² -x + 2 (-1, 5)

2) y = x² + x + 1 at (-2, 3)

Find the slope of the curve at the given point. Show your solution.

1) y = 4 - 3x² at (-1, 1)

2) y = x² – 5 at (1, -4)

3) y = 2x² – 5 at (2, 3)

A cylindrical container has one end opened and the other end closed. It has a circular base of radius r cm. Given that the total surface area of the container is 200π cm2.

(a) Show that the volume of the container is V = 100πr - πr^3/2 cm3

(b) Find the maximum value of V.

b) Find whether the following series are convergent or divergent √ 𝟏 /𝟒 + √ 𝟐 /𝟔 + √ 𝟑 /8 + ...

Find the dimensions of a right circular cone of minimum volume V that can be

circumscribed about a sphere of radius 8 inches.

Find the 𝑛th order differential coefficient of 𝑦 = sin 𝑥 log(𝑎𝑥 + 𝑏).

1. Two given lines are either parallel, skew or intersecting. (6 marks)
2. Determine, if there is one, the point of intersection of the lines given by the equations

x−1/−3=y−8/6=z−3/−2    and    x−8/4=y+3/−5=z−7/2

    b. Give the equations of two lines that meet at the point (–1 5, 2) and which meet at right angles, but do not use that point in either of the equations. Explain your reasoning

The equation for the instantaneous voltage across a discharging capacitor is given by 𝑣 = 𝑉𝑂𝑒 − 𝑡 𝜏 , where 𝑉𝑂 is the initial voltage and 𝜏 is the time constant of the circuit. The tasks are to: a) Draw a graph of voltage against time for 𝑉𝑂 = 12𝑉 and 𝜏 = 2𝑠, between 𝑡 = 0𝑠 and 𝑡 = 10𝑠. b) Calculate the gradient at 𝑡 = 2𝑠 and 𝑡 = 4𝑠. c) Differentiate 𝑣 = 12𝑒 − 𝑡 2 and calculate the value of 𝑑𝑣 𝑑𝑡 at 𝑡 = 2𝑠 and 𝑡 = 4𝑠. d) Compare your answers for part b and part c. e) Calculate the second derivative of the instantaneous voltage ( 𝑑 2 𝑣 𝑑𝑡2 ).

Find the position vector when acceleration a(t)= 2i + j + 3k

(A) A company's total revenues (in millions of dollars) t years from

now are predicted by R(t). Briefly explain what R(8) = 126.60

and R' (8) = 25.30 mean in the context of this question.