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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
Show that the MacLaurin's series for f (z) =sin z is given by
z z 3 s "' (- I)" z211+J
f(z) =z--+- ....+L ( ) +...
Evaluate each of the following limits. Simplify all answers completely. Show all work toward your answer or provide an explanation for how you arrived at your answer.
1) lim x--> (x^2 -1)/(x^2 -8x+7)
2) lim x-->0^negative(-) (2)/(tan(x))
3) lim x--> infinity (cos(t))/(e^3t)
4) lim x--> (-)infinity (-3x^3 +5x^2 -4)
Find each of the following derivatives. Simplify all answers completely. Show all work toward your answer or provide an explanation for how you arrived at your answer.
1) f(x)=sqroot e^2x + 8x^2 e^x
2) g(x)= xsin(x)/1+cos(x)
3) h(x)=e^-x sin(x)
4) y=sec(x)tan(x)
5) f(t)= ((t-1)(2t^2 -1))/(t^3 -1)
A cone is generated when the region is bounded by the line x=y and the vertical line x=0 and x=r is rotated about the x-axis . Use the pappus theorem to show that , Surface area of the cone is given by S=β2ΟrΒ²
