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answers, in the form of a short proof or a counter example.
(i) The function f defined by f(x) = tan(2x) is a periodic function with period Ο.
(ii) The function f: R---> R defined by f(x) = 1-|x| is differentiable at x=1
(iii) The function f: [3,4] ----> R defined by
f(x) = x
x^2+y^2= 4x
and
y^2=x
above x-axis
A spherical balloon is deflated so that its volume is decreasing at a rate of 3 ftΰ¬·/min. How fast is the diameter of the balloon decreasing when the radius is 2 ft?Β
Let π be a function which is everywhere differentiable and for which π(2) = β3 and π β² (π₯) = βπ₯ 2 + 5. Given that π is defined such that π(π₯) = π₯ 2π ( π₯ π₯ β 1 ), show that π β² (2) = β24.
Differentiate the following functions (i) If π¦ = π β3π‘ sin 4π‘, prove that π 2π¦ ππ₯ 2 + 6 ππ¦ ππ₯ + 2π¦ = 0. (ii) Given that sin(π₯ 2 + π¦) = π¦ 2 (3π₯ + 1), show that ππ¦ ππ₯ = 2π₯ cos(π₯ 2 + π¦) β 3π¦ 2 2π¦(3π₯ + 1) β cos(π₯ 2 + π¦
Differentiate the following functions (i) π(π₯) = π₯π π₯ ππ ππ₯, (ii) β(π₯) = ln(π₯ + βπ₯ 2 β 1), (iii) π¦ = cos(π βπ‘ππ 3π₯ ), (iv) β1 + sin3(π₯π¦ 2) = π¦, (v) π¦(π₯) = (logπ π₯) π₯ tan(π π₯ ), (vi) π(π₯) = sinβ1 (β1 β 9π₯ 2).Β
Use an appropriate local linear approximation to estimate 3β9
The volume of a sphere is to be computed from the measured value of its radius. Estimate the maximum permissible percentage error in the measurement if the percentage error in the volume must be kept within Β±6%.
A spherical balloon is deflated so that its volume is decreasing at a rate of 3 ftΰ¬·/min. How fast is the diameter of the balloon decreasing when the radius is 2 ft?Β
P=25000-2q and cost of production is
Tc=2000000-20000q +5q2
Where q :number of units sold
P:price
Tc : total cost
How many units should be produced to maximize profit and what is the maximum profit
