Estimate the derivative at the given point by using a calculator.
Problem 1 :
f(x) = x√(2 - x); find f^{'}(-10).
Solution :
f(x) = x√(2 - x)
Using product rule.
d/dx[f ⋅ g] = f^{'}g + fg^{'}
By using a calculator.
f^{'}(-10) = 4.907
Problem 2 :
f(x) = sec(5x); find f^{'}(2).
Solution :
f(x) = sec(5x)
Using product rule.
d/dx[f ⋅ g] = f^{'}g + fg^{'}
f'(x) = sec (5x)tan(5x) ⋅ 5
f'(x) = 5 sec 5x tan 5x
To find f^{'}(2) :
f'(2) = 5 sec 5 ⋅ 2 tan 5 ⋅ 2
= 5 sec 10 tan 10
= 5 ⋅ (1/cos 10) tan 10
By using a calculator.
f^{'}(2) = -3.863
Problem 3 :
f(x) = In(√x); find f^{'}(1).
Solution :
f(x) = In(√x)
Problem 4 :
Solution :
Problem 5 :
f(x) = tan(sin x); find f^{'}(-3)
Solution :
Given, f(x) = tan(sin x)
Using product rule.
d/dx[f ⋅ g] = f^{'}g + fg^{'}
f^{'}(x) = sec^{2}(sin x) cos x
f^{'}(-3) = sec^{2}(sin (-3)) cos (-3)
f^{'}(-3) = sec^{2}(sin 3) (cos 3)
= (1 + tan^{2}(sin 3)) (cos 3)
= (1 + tan^{2} 0.14112) × (-0.989992)
= sec^{2 }0.14112 × (-0.989992)
= - sec^{2 }0.14112 × 0.989992
= -(1/cos 0.14112)^{2} × 0.989992
By using a calculator.
f^{'}(-3) = -1.009
Problem 6 :
f(x) = 2^{In(x)}; find f^{'}(2).
Solution :
Problem 7 :
The model f(t) = x/cos x measures the height of bird in meters where t is seconds. Find f^{'}(2).
Solution :
f^{'}(2) = 8.098 m/sec
Problem 8 :
The model f(t) = sin^{2}(t) measures the depth of a submarine measured in feet where t is minutes. Find f^{'}(12.5).
Solution :
Given, f(t) = sin^{2}(t)
f'(t) = 2 (sin t) (cos t)
To find f'(12.5) :
f'(12.5) = 2 (sin 12.5) (cos 12.5)
= 2 (-0.066) (0.9978)
= -0.132 × 0.9978
f'(12.5) = -0.13 ft/min
Problem 9 :
The model f(t) = √x - 1/(x - 1) measures the number of stocks sold where t is days. Find f'(12).
Solution :
f'(12) = 0.1526 stocks/day
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM