I am a MBA graduate from a reputed university and want to make mark in teaching.
If $ f(x) = \frac{x^2-x}{x-1} $ and $ g(x) = x $ is it true that $ f = g $?
The point $ P(2, -1) $ lies on the curve $ y = 1/(1-x) $.
(a) If $ Q $ is the point $ (x, 1/(1-x)) $, use your calculator to find the slope of the secant line $ PQ $ (correct to six decimal places) for the following values of $ x $: (i) $ 1.5 $ (ii) $ 1.9 $ (iii) $ 1.99 $ (iv) $ 1.999 $ (v) $ 2.5 $ (vi) $ 2.1 $ (vii) $ 2.01 $ (viii) $ 2.001 $
(b) Using the results of part (a), guess the value of the slope of the tangent line to the curve at $ P(2, -1) $.
(c) Using the slope from part (b), find an equation of the tangent line to the curve at $ P(2, -1) $.
(a) If $ g(x) = 2x + 1 $ and $ h(x) = 4x^2 + 4x + 7 $, find a function $ f $ such that $ f \circ g = h $. (Think about what operations you would have to perform on the formula for $ g $ to end up with the formula for $ h $.)(b) If $ f(x) = 3x + 5 $ and $ h(x) = 3x^2 + 3x + 2 $, find a function $ g $ such that $ f \circ g = h $.
The $ signum $ (or sign) $ function$ , denoted by sgn, is defined by sgn $ x = \left\{ \begin{array}{ll} -1 & \mbox{if $ x < 0 $}\\ 0 & \mbox{if $ x = 0 $}\\ 1 & \mbox{if $ x > 0 $} \end{array} \right.$
(a) Sketch the graph of this function.(b) Find each of the following limits or explain why it does not exist. (i) $ \displaystyle \lim_{x \to 0^+}\text{sgn $ x $} $ (ii) $ \displaystyle \lim_{x \to 0^-}\text{sgn $ x $} $ (iii) $ \displaystyle \lim_{x \to 0}\text{sgn $ x $} $ (iv) $ \displaystyle \lim_{x \to 0}| \text{sgn $ x $} | $
(a) From the graph of $ f $, state the numbers at which $ f $ is discontinuous and explain why.(b) For each of the numbers stated in part (a), determine whether $ f $ is continuous from the right, or from the left, nor neither.
The gravitational force exerted by the planet Earth on a unit mass at a distance $ r $ from the center of the planet is
$ F(r) = \left\{ \begin{array}{ll} \frac{GMr}{R^3} & \mbox{if $ r < R $}\\ \frac{GM}{r^2} & \mbox{if $ r \ge R $} \end{array} \right.$
where $ M $ is the mass of Earth, $ R $ is its radius, and $ G $ is the gravitational constant. Is $ F $ a continuous function of $ r $?
find the area of the shaded region y=x , y=x^2
A 100 N block of metal hangs in equilibrium and is suspended bytwo cables, as shown in the figure. The tensions in the cables areclosest toThe figure shows a block hanging from two cables that areconnected to the ceiling. The cables each make an angle of 37°relative to the ceiling such that an isosceles triangle isformed.
A piece of wire 13 m long is cut into two pieces. One piece is bent into the shape of a circle of radius ð‘Ÿ and the other is bent into a square of side ð‘ . How should the wire be cut so that the total area enclosed is:a) a maximum? ð‘Ÿ= and ð‘ = .b) a minimum? ð‘Ÿ= and ð‘ =.
1A) Gravel is being dumped from a conveyor belt at a rateof 35 ft3/min, and its coarseness is such thatit forms a pile in the shape of a cone whose base diameter andheight are always equal. How fast is the height of the pileincreasing when the pile is 5 ft high? (Round your answerto two decimal places.)1B)Each side of a square is increasing at a rateof 6 cm/s. At what rate is the area of the squareincreasing when the area of the squareis 36 cm2?1C)Two sides of a triangle are 5 m and 8 m inlength and the angle between them is increasing at a rate of 0.06rad/s. Find the rate at which the area of the triangle isincreasing when the angle between the sides of fixed length is pi/3rad? m2/s1D)If a snowball melts so that its surface area decreases at a rateof 7 cm2/min, find the rate at which thediameter decreases when the diameter is 10 cm. cm/min
A tech company determines the total cost, in dollars, of producing x hundred units of a computer to be C(x) = 3000 + 2000x, and the total profit to be P(x) = -200x^2 + 98000x - 3000. (a) Find the marginal average revenue. (b) What price per unit yields the maximum revenue?
A packaging company has been commissioned toproduce closed cardboard boxes. The boxes are tohave a square base and the required volume of the boxesis 512 m^3. The company would like to use as littlematerial as possible in order to minimize production costs. Whatdimensions should the boxes have?
