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g−1(t)=
where d is describe by 0 ≤ x ≤ 1 , 0 ≤ y ≤ x
where d is describe by 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1
Differentiate the Inverse Trigonometric functions:
1. Find 𝑓′′(𝜋) if 𝑓(𝑥) = 𝑎𝑟𝑐𝑠𝑖𝑛(𝑐𝑜𝑠𝑥).
2. If 𝑐𝑜𝑡^−1(𝑥𝑦) − 𝑡𝑎𝑛^−1(𝑦/𝑥) = 0, find 𝑑𝑦/𝑑𝑥.
3. ℎ(𝑡) = 𝑎𝑟𝑐𝑐𝑜𝑠 (𝑡−1/𝑡+1).
Differentiate the Logarithmic functions:
𝑦 = 𝑥^3 𝑙𝑛^2 𝑥 + 𝑥^2 𝑙𝑛𝑥^3 + 1
Differentiate the Logarithmic functions:
1. 𝑦 = 𝑥^3𝑙𝑛^2𝑥 + 𝑥^2𝑙𝑛𝑥^3 + 1
2. 𝑔(𝑥) = 𝑙𝑛 (𝑥^2+1/𝑥^3+5)
3. ℎ(𝑥) = 𝑙𝑜𝑔(𝑐𝑜𝑡4𝑥 − 𝑐𝑠𝑐4𝑥)
4. 𝑦 = 𝑙𝑛(𝑥^3𝑠𝑖𝑛2𝑥)
5. 𝑦 = 𝑙𝑛(𝑐𝑜𝑠5𝑥 + 𝑠𝑖𝑛5𝑥)
A company has learned that when it initiates a new sales campaign, the number of sales per
day increases. However, the number of extra daily sales per day decreases as the impact of the campaign wears off. For a specific campaign the company has determined that if there are e 𝑆(𝑡) extra daily sales as a result of the campaign and 𝑡 days have elapsed since the campaign ended, then 𝑆(𝑡) = 1000 (3^−𝑡/2). Find the rate at which the extra daily sales are decreasing when (a) 𝑡 = 4 and (b) 𝑡 = 10.
In a telegraph cable, the measure of the speed of the signal is proportional to 𝑥^2 𝑙𝑛 (1/𝑥), where
𝑥 is the ratio of the measure of the radius of the core of the cable to the measure of the thickness of the cable’s winding. Find the value of 𝑙𝑛𝑥 for which the speed of the signal is greatest.
A particular company has determined that when its weekly advertising expense is 𝑥 dollars,
then if 𝑆 dollars is its total weekly income from sales, 𝑆 = 4000 𝑙𝑛𝑥.
(a) Determine the rate of change of sales income with respect to advertising expense when $800 is the weekly advertising budget.
(b) If the weekly advertising budget is increased to $950 dollars, what is the approximate increase in the total weekly income from sales?
