Math 190 Chapter 3 Lecture Notes. Professor Miguel Ornelas

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1 Math 190 Chapter 3 Lecture Notes Professor Miguel Ornelas 1

2 M. Ornelas Math 190 Lecture Notes Section 3.1 Section 3.1 Derivatives of Polynomials an Exponential Functions Derivative of a Constant Function x (c) = 0 The Power Rule In n is any real number, then x (xn ) = nx n 1 The Constant Multiple Rule If c is a constant an f is a ifferentiable function, then x [cf(x)] = c x f(x) The Sum Rule If f an g are both ifferentiable, then [f(x) + g(x)] = x x f(x) + x g(x) The Difference Rule If f an g are both ifferentiable, then [f(x) g(x)] = x x f(x) x g(x) Derivative of the Natural Exponential Function x (ex ) = e x Differentiate the function. a. f(x) = 3 7 b. g(x) = 7 4 x2 3x + 12 Section 3.1 continue on next page... 2

3 M. Ornelas Math 190 Lecture Notes Section 3.1 (continue). h(t) = 4 t 4e t e. S(R) = 4πR 2 x + x f. y = x 2 g. F (z) = A + Bz + Cz2 z 2 h. k(r) = e r + r e i. y = e x Fin the equation of the tangent line to the curve at the given point. y = x x, (1, 0) Section 3.1 continue on next page... 3

4 M. Ornelas Math 190 Lecture Notes Section 3.1 (continue) The equation of motion of a particle is s = t 4 2t 3 + t 2 t, where s is in meters an t is in secons. (a) Fin the velocity an acceleration as functions of t. (b) Fin the acceleration after 1 s. Fin an equation of the tangent line to the curve y = x that is parallel to the line 32x y = 15. 4

5 M. Ornelas Math 190 Lecture Notes Section 3.2 Section 3.2 The Prouct an Quotient Rules The Prouct Rule If f an g are both ifferentiable, then [f(x)g(x)] = f(x) [g(x)] + g(x) x x x [f(x)] The Quotient Rule If f an g are ifferentiable, then x [ ] f(x) g(x) [f(x)] f(x) = x x [g(x)] g(x) [g(x)] 2 Differentiation Formulas x (c) = 0 x (xn ) = nx n 1 x (ex ) = e x (cf) = cf (f + g) = f + g (f g) = f g (fg) = fg + gf ( f g ) = gf fg g 2 Differentiate. a. g(x) = (x + 2 x)e x b. G(x) = x2 2 2x + 1 Section 3.2 continue on next page... 5

6 M. Ornelas Math 190 Lecture Notes Section 3.2 (continue) c. y = 1 t 3 + 2t 2 1. h(r) = aer b + e r Fin an equation of the tangent line to the given curve at the specifie point. y = 1 + x 1 + e x, ( 0, 1 ) 2 If f(2) = 10 an f (x) = x 2 f(x) for all x, fin f (2). Section 3.2 continue on next page... 6

7 M. Ornelas Math 190 Lecture Notes Section 3.2 (continue) Section 3.3 Derivatives of Trigonometric Functions Limits sin θ cos θ 1 lim = 1 lim = 0 θ 0 θ θ 0 θ Derivatives of Trigonometric Functions (sin x) = cos x x (csc x) = csc x cot x x (cos x) = sin x x (sec x) = sec x tan x x x (tan x) = sec2 x x (cot x) = csc2 x Differentiate. a. f(x) = x cos x + 2 tan x b. f(t) = cot t e t c. y = cos x 1 sin x. f(t) = te t cot t Section 3.3 continue on next page... 7

8 M. Ornelas Math 190 Lecture Notes Section 3.3 (continue) Fin an equation of the tangent line to the curve y = 3x + 6 cos x at the point (π/3, π + 3). A laer 10 ft long rests against a vertical wall. Let θ be the angle between the top of the laer an the wall an let x be the istance from the bottom of the laer to the wall. If the bottom of the laer slies away from the wall, how fast oes x change with respect to θ when θ = π/3? Fin the limit. cos θ 1 lim θ 0 sin θ sin(x 1) lim x 1 x 2 + x 2 Section 3.3 continue on next page... 8

9 M. Ornelas Math 190 Lecture Notes Section 3.3 (continue) Section 3.4 The Chain Rule The Chain Rule If g is ifferentiable at x an f is ifferentiable at g(x), then the composite function F = f g efine by F (x) = f(g(x)) is ifferentiable at x an F is given by the prouct F (x) = f (g(x)) g (x) In Leibniz notation, if y = f(u) an u = g(x) are both ifferentiable functions, then y x = y u u x The Power Rule Combine with the Chain Rule If n is any real number an u = g(x) is ifferentiable, then x (un n 1 u ) = nu x or x [g(x)]n = n [g(x)] n 1 g (x) Fin the erivative of the function. a. F (x) = (1 + x + x 2 ) 99 b. f(t) = t sin πt c. f(t) = 2 t3. s(t) = 1 + sin t 1 + cos t Section 3.4 continue on next page... 9

10 M. Ornelas Math 190 Lecture Notes Section 3.4 (continue) e. y = x 2 e 1/x f. y = [ x + (x + sin 2 x) 3] 4 Fin the 1000th erivative of f(x) = xe x. The average bloo alcohol concentration BAC of eight male subjects was measure after consumption of 15 ml of ethanol (corresponing to one alcoholic rink). The resulting ata were moele by the concentration function C(t) = te t where t is measure in minutes after consumption an C is measure in mg/ml. (a) How rapily was the BAC increasing after 10 minutes? (b) How rapily was it ecreasing half an hour later? Section 3.4 continue on next page... 10

11 M. Ornelas Math 190 Lecture Notes Section 3.4 (continue) Section 3.5 Implicit Differentiation Derivatives of Inverse Trigonometric Functions 1 x (sin 1 x) = 1 x 2 ( csc 1 x ) 1 = x x x 2 1 x ( cos 1 x ) 1 = 1 x 2 ( sec 1 x ) = x 1 x x 2 1 ( tan 1 x ) = 1 ( cot 1 x 1 + x 2 x ) = 1 x 1 + x 2 Fin the erivative of the function. a. 2x 2 + xy y 2 = 2 b. xe y = x y c. e y sin x = x + xy. x sin y + y sin x = 1 Section 3.5 continue on next page... 11

12 M. Ornelas Math 190 Lecture Notes Section 3.5 (continue) Use implicit ifferentiation to fin an equation of the tangent line to the curve at the given point. x 2 + 2xy + 4y 2 = 12, (2, 1) (ellipse) Fin the the erivative of the function. Simplify where possible. a. y = cos 1 ( sin 1 t ) b. y = tan 1 ( x 1 + x 2 ) 12

13 M. Ornelas Math 190 Lecture Notes Section 3.6 Section 3.6 Derivatives of Logarithmic Functions Derivatives of Logarithmic Functions x (log b x) = 1 x ln b x (ln x) = 1 x x (ln u) = 1 u u x x ln x = 1 x Differentiate the function. a. f(x) = ln(sin 2 x) ( b. h(x) = ln x + ) x 2 1 c. P (v) = ln v 1 v. y = ln ( e x + xe x) Section 3.6 continue on next page... 13

14 M. Ornelas Math 190 Lecture Notes Section 3.6 (continue) Fin an equation of the tangent line to the curve at the given point. y = x 2 ln x, (1, 0) Use logarithmic ifferentiation to fin the erivative of the function. a. y = e x cos 2 x x 2 + x + 1 b. y = (sin x) ln x 14

15 M. Ornelas Math 190 Lecture Notes Section 3.7 Section 3.7 Rates of Change in the Natural an Social Sciences If a ball is thrown vertically upwar with a velocity of 80 ft/s, then its height after t secons is s = 80t 16t 2. (a) What is the maximum height reache by the ball? (b) What is the velocity of the ball when it is 96 ft above the groun on its way up? On its way own? If a tank hols 5000 gallons of water, which rains from the bottom of the tank in 40 minutes, the volume V of water remaining in the tank after t minutes is ( V = ) 2 40 t 0 t 40 Fin the rate at which water is raining from the tank after (a) 5 min, (b) 10 min, (c) 20 min, an () 40 min. At what time is the water flowing out the fastest? The slowest? Section 3.7 continue on next page... 15

16 M. Ornelas Math 190 Lecture Notes Section 3.7 (continue) Newton s Law of Gravitation says that the magnitue F of the force exerte by a boy of mass m on a boy of mass M is F = GmM r 2 where G is the gravitational constant an r is the istance between the boies. (a) Fin F/r an explain its meaning. What oes the minus sign inicate? (b) Suppose it is known that the earth attracts an object with a force that ecreases at the rate of 2 N/km when r = 20, 000 km. How fast oes this force change when r = 10, 000 km? 16

17 M. Ornelas Math 190 Lecture Notes Section 3.8 Section 3.9 Relate Rates The length of a rectangle is increasing at a rate of 8 cm/s an its with is increasing at a rate of 3 cm/s. When the length is 20 cm an the with is 10 cm, how fast is the area of the rectangle increasing? If a snowball melts so that its surface area ecreases at a rate of 1 cm 2 /min, fin the rate at which the iameter ecreases when the iameter is 10 cm. A laer 10 ft long rests against a vertical wall. If the bottom of the laer slies away from the wall at a rate of 1 ft/s, how fast is the angle between the laer an the groun changing when the bottom of the laer is 6 ft from the wall? Section 3.9 continue on next page... 17

18 M. Ornelas Math 190 Lecture Notes Section 3.9 (continue) A baseball iamon is a square with sie 90 ft. A batter hits the ball an runs towar first base with a spee of 24 ft/s. (a) At what rate is his istance from secon base ecreasing when he is halfway to first base? (b) At what rate is his istance from thir base increasing at the same moment? A spotlight on the groun shines on a wall 12 m away. If a man 2 m tall walks from the spotlight towar the builing at a spee of 1.6 m/s, how fast is the length of his shaow on the builing ecreasing when he is 4 m from the builing? Section 3.9 continue on next page... 18

19 M. Ornelas Math 190 Lecture Notes Section 3.9 (continue) Section 3.10 Linear Approximations an Differentials Linearization of f at a f(x) f(a) + f (a)(x a) L(x) = f(a) + f (a)(x a) Fin the linearization L(x) of the function at a. f(x) = x 3 x 2 + 3, a = 2 Differentials y = f (x)x y = f(x + x) f(x) f(a + x) f(a) + y Section 3.10 continue on next page... 19

20 M. Ornelas Math 190 Lecture Notes Section 3.10 (continue) Fin the ifferential of the function. y = 1 + 2u 1 + 3u Use a linear approximation (or ifferentials) to estimate Use a linear approximation (or ifferentials) to estimate