1 MATH 131 Exam 2 , f x is negative Monday, October 19th 2.6, 2.7, 2.8, 3.1, 3.2, 3.3, 3.4, 3.7, 3.8, 3.9 Antiderivatives of f is a function F , such that • 2.6 – Derivatives and Rates of Change Tangent to a Curve y f x at x a is y f a m x a F f • 3.1 – Derivatives y x n , y n x n 1 y c , a constant, y 0 Power rule: Constant multiple rule: y ax n , then y m x m a f a f is concave down y a n x n 1 f a h f a f a h 0 h f a h f a s average velocity t h f a h f a velocity v a lim h 0 h f a h f a derivative f a lim h 0 h y f x2 f x1 rates of change: x x2 x1 m lim marginal cost: C x Sum rule: h x f x g x , h x f x g x Difference rule: h x f x g x y e x , y e x y a x , y a x ln a • 3.2 – Product and Quotient Rules • 2.7 – The Derivative As a Function h x f x g x , y f g , y f g g f f g f f g y , y g g2 • 3.3 – Derivatives of Trig Functions f x 0 when f has horizontal tangents f x is positive f is increasing y sin x , y cos x y cos x , y sin x f x is negative f is decreasing y tan x , y sec2 x dy f x y dx d2y f x y 2 dx If f is differentiable at xa Important properties 1 1 1 csc x, sec x, cot x sin x cos x tan x sin 2 A cos 2 A 1 sin x tan x cos x , f is continuous at Continuous functions are not differentiable at: a. hole or gap b. vertical asymptote c. corner or cusp • 2.8 - What Does f say about f ? f x is positive f is increasing f x is negative f is decreasing f x is positive f is concave up • 3.4 – The Chain Rule d n du u n u n 1 dx dx 2 mess y mess , ye y a mess , y a mess mess ln a y n mess n mess , y e mess n 1 mess y f g x f g x y f g x g x dy dy du dx du dx • 3.7 – Derivatives of Logarithms 1 y ln x , y x 1 x ln b mess y ln mess , y mess mess y logb mess , y mess ln b f x g x y ln h x y ln f x ln g x ln h x y log b x , y y • 3.8 – Rates of Change In the Natural & Social Sciences Average rate of change = Instantaneous rate of change = Linear approximation L x f a f a x a aka: Tangent line approximation Differentials dy f x dx Relative error: using differentials If V volume dV instantaneous rate of change of volume dr dV used to calculate maximum error f g h f g h f x g x h x ln y ln f x g x h x ln y ln f x ln g x ln h x differentiate y f g h y f g h f g h y y g h f y dy dx • 3.9 – Linear Approximations and Differentials Logarithmic Differentiation y y x f g f g h h f g h .. 3 Sample Problems : 1. F x 8x , find F 3 3 x2 2. Write the equation of the tangent to the curve F x 8x at x 3. 3 x2 3. Find f a given f t 5t 11 t4 4. The quantity (in pounds) of dog food sold by Alpo, at a price of p dollars per pound, is Q f p . 9. f x x 4 a. What is the domain of the function. b. f x c. What is the domain of the derivative? a. What is the meaning of f 5 ? 10. Given the graph of y f x , draw the b. What are the units of f 5 ? c. In general, will f 5 be positive or graph of f x . negative? 5. If a rock is thrown vertically upward on the planet Magrathea, with a velocity of 20m/s, its height in meters after seconds is given by H 20t 2.5t 2 . a. When will the rock hit the surface? b. With what velocity will the rock hit 11. Given the graph of f x , the surface? 6. Given f x h f x x2 x h x2 x h 2 2 , Find the slope of the tangent to a. Over what intervals is f x ? b. At what x values does f x have a minimum? y f x at x 1. 7. Matching: The graphs of 3 derivatives are shown. Match the graph of each function with the graph of its derivative. 8. The graph of f x is shown. State the x values at which f is not differentiable. 12. Let f t represent the temperature of your patient at time t. f 2 99.5 F , f 2 1.5 , f 2 0.5 . Explain. c Marcia Drost, October 19, 2015 4 13. Sketch a function whose 1st and 2nd b. g x x 9 2 x 7 c. h x d. b f x ax 2 c x e. g x x x 4 f. h x derivatives are always negative. 5 x3 4 x2 2 x 1 x 17. The equation of motion is s t 3 9t , 14. f 1 f 1 0, a. v t f x 0, if x 1, b. a t f x 0 if 1 x 2, c. Find the acceleration when the f x 1, if x 2 , velocity is zero. f x 0 if 2 x 0, 18. Find the 1st and 2nd derivatives of inflection point: 0,1 f x e2 x x3 19. Find the points on the curve where the tangent line is horizontal when y x3 24 x 2 . 20. Differentiate: a. f x 3e 2 x x 15. Given f x x e x 1 8 3 x 5x2 2 x x b. g x 2 a. On what interval is f x ? b. On what interval is f x ? c. h x x3 2 x2 d. f x x 2 3x e r 21. f 3 1, g 3 2, f 3 4, g 3 3 Find h 3 . a. h x 4 f x 5 g x b. h x 16. Find the derivative: a. 3 2 f x x 4 x 3 4 x e3 2 3 f x 1 g x c. h x f g x 5 22. Differentiate: a. y sin x cos x 1 b. y 2 xe x sin x x f g f’ g‘ 2 3 4 5 -1 3 5 2 -2 6 4 -1 3 3 3 c. y e cos x 2x 2 23. Write the equation of the tangent to the curve y 4 x 2cos x at x 0. 24. Write the equation of the tangent line to 8 at x 0. y sin x 2 cos x cos x 25. f 60 2, f 60 3, g x f x Find g 60 26. An elastic band is hung on a hook and a mass is hung on the lower end of the band. When the mass is pulled down and released, it vibrates vertically. The equation of motion s , in cm, and t 0 in sec, is given by s 4cos t 5sin t . Find the velocity and acceleration at time . 27. Find the derivative of: a. y 2 x 3 3 4 b. y sin 2 a3 x 2 29. h x f c. y log 2 x 2 12 x d. y ln x3 e2 x x e 3 x 5 x 5 e. y ln 2 3 x 9 f. y ln 1 e 2 x g. y 3 ln 2x x2 31. A particle moves according to the law of motion s f t 3t 3 24t 2 72t t 0, where is measured in seconds and s in feet. b. When is the particle at rest? c. What is the total distance traveled in 32. Given the graph of the velocity function, 3 28. Find the equation of the tangent to the curve y b. y ln 50sin 2 x the first 4 sec? 8 xcos x x2 2 e. y x5 a. y sin 5ln x a. Find the velocity at 3 sec. c. y 2 x e kx d. y e 30. Differentitate: 4 at x 0. , 1 e x g x Find h 4 . in sec, when is the article speeding up? 6 33. If a ball is thrown vertically upward with a velocity of 64ft/sec, then its height after sec is s 64t 16t 2 . a. What is the maximum height reached? b. What is the velocity of the ball when it first reaches 50 feet? 34. Find the linearization L x of the function f x x3 5 x 2 at a 1 . 35. Use linear approximation to estimate the value of 15.9 . 36. Find the differential of y u4 u 3