Calculus Answers

4. (Section 4.5) Consider the R 2 − R function f defined by f (x, y) = x 2 + y y . Determine each of the following limits, if it exists. (a) lim (x,y)→C1 (0,0) f (x, y), where C1 is the curve y = x. (2) (b) lim (x,y)→C2 (0,0) f (x, y), where C2 is the curve y = 2x. (2) (c) lim (x,y)→C3 (0,0) f (x, y), where C3 is the curve y = x 2 . (2) (d) lim (x,y)→(0,0) f (x, y).

2. Consider the R − R 2 function r defined by r (t) =    (t, t2 ) if t ∈ [−2, 0] (t, t) if t ∈ (0, 2) t, t2 if t ∈ [2, 3] (a) Write down the domain of r. (1) (b) Is r continuous at t = 0? (2) (c) Is r continuous at t = 2? (2) (d) Sketch the curve r.

find the area in the second quadrant bounded by the curve 2x^2+4x+y=0?

Let

f

be a differentiable function on

[Alpha, beta ]

and

x belongs to [alpha, beta ].

Show that, if

f '(x) = 0

and

f ''(x) =0,

then

f

must have a local maximum at

x.

If (𝜑)𝑥, 𝑦, 𝑧) = 𝑥𝑦 2 𝑧 and 𝐴 = 𝑥𝑧𝑖 + 𝑥𝑦 2 𝑗 + 𝑦𝑧 2𝑘, find 𝜕 3 𝜕2𝑥𝜕𝑧 𝜑𝐴 at point 2, −1,1 .

1. A baseball diamond has the shape of a square with sides 90 ft long. A player 60 ft from second base is running towards third base at a speed of 28 ft/min. At what rate is the player’s distance from the home plate changing?
2. A ladder inclined at 60 ° with the horizontal is leaning against a vertical wall. The foot of the ladder is 3 meters away from the foot of the wall. A boy climbs the ladder such that his distance z meters with respect to the foot of the ladder is given by z=6t, where t is the time in seconds. Find the rate at which his vertical distance from the ground changes with respect to t. Find the rate at which his distance from the foot of the wall is changing with respect to t when he is 3 meter away from the foot of the ladder.

1. Starting from the same point, Reden starts walking eastward at 60 cm/s while Neil starts running towards the south at 80 cm/s. How fast is the distance between Reden and Neil increasing after 2 seconds?
2. A balloon, in the shape of a right circular cylinder, is being inflated in such a way that the radius and height are both increasing at the rate of 3 cm/s and 8 cm/s, respectively. What is the rate of change of its total surface area when its radius and height are 60 cm and 140 cm, respectively?

Integrate g(x, y) = x⁴ + y² over the region 

bounded by y = x, y = 2x and x = 2. 

 Find the minimum value of the function 

f(x, y) = x2 + 2y2 on the circle x2 + y2 = 1. 

Evaluate   where   

F(x,y,z)=xzi−yzk

  and c is the line segment  from (3,0,1) to (-1,2,0)