Section Derivatives of Polynomials and Exponential Functions

Transcription

1 Section Derivatives of Polynomials and Exponential Functions Constant Function Rule: If f(x)=c where c is a constant then Power Rule: If f(x)=x n, where n is a real number, then Constant Multiple Property: If f(x)=k g(x), where k is any real number, then Sum and Difference Property: If h(x)= f(x)±g(x) where f and g are both differentiable functions, then Example 1:Differentiate the following functions: a) f(x)=3 b) f(x)=e c) f(x)= 5 8 d) f(x)=x 4 e) f(x)=x 1/2 f) f(x)= 3 x 5 g) f(x)= 5 x 2 h) y=3x 2 + 7x 9 1

2 i) m(t)= 5t t 1/2 + e j) k(x)= x3 + 4x 2 17+x x k) y= 3x2 + x 4 5 x Example 2: If a book is dropped from a building 400 feet tall, its height above the ground (in feet) after t seconds is given by s(t)=400 16t 2 a) Compute s (t) and interpret. b) Compute s(2) and s (2) and interpret. c) When does the book hit the ground? d) What is the impact velocity? 2

3 Example 3: If f(x)=3x 4 2x 2, where does the graph of the function have a horizontal tangent line? Derivatives of Exponential Functions If f(x)=b x then If f(x)=e x then Example 4: Find the derivative of each of the following functions: a) f(x)=2e x b) f(x)=7(5) x Example 5: The resale value R (in dollars) of a company car after t years is estimated to be given by R(t)= 20000(0.86) t What is the rate of depreciation (in dollars per year) after 1 year? 3 years? Section 3.1 Highly Suggested Homework Problems: 3, 7, 11, 15, 19, 23, 27, 31, 41, 45, 47, 49, 53, 65 3

4 Section The Product and Quotient Rules Product Rule: If h(x)= f(x) g(x) and if f (x) and g (x) exists, then Quotient Rule: If h(x)= f(x) g(x) and if f (x) and g (x) exist, then Example 1: Find the derivative of the following functions a) f(x)=x 2 (x 2 + 4x) b) g(x)= x x c) h(x)=(x 2 + 3)( 4 x+ 8 x 3 ) d) k(x)= 3 x+7x x 2 4x+ 1 x 4

5 e) F(x)=5x 4 e x f) g(x)= x2 e x e x Example 2: Find the equation of the tangent line to f(x)= x x 3 4x 2 at x=

6 Example 3: Suppose that f(2)= 1, g(2)=3, f (2)= 4, and g (2)=6. Find h (2) for each of the following: a) h(x)=2 f(x) 3g(x) b) h(x)= f(x)g(x) c) h(x)= f(x) g(x) d) h(x)= f(x) 1+g(x) Example 4: Let P(x)=F(x)G(x) and Q(x)=F(x)/G(x), where F and G are the functions whose graphs are shown. a) Find P (2) b) Find Q (7) Section 3.2 Highly Suggested Homework Problems: 1, 5, 9, 13, 17, 21, 25, 27, 29, 39, 41, 43, 45, 47 (do not simplify) 6

7 Section Derivatives of Trigonometric Functions Derivatives of Trigonometric Functions: If f(x)=sinx, then f (x)=cosx If f(x)=cosx, then f (x)= sinx If f(x)=tanx, then f (x)= If f(x)=cscx, then f (x)= If f(x)=secx, then f (x)= If f(x)=cotx, then f (x)= Example 1: Determine the missing derivatives of the trigonometric functions above. 7

8 Example 2: Differentiate the following functions: a) f(x)= xsinx b) g(θ)=e θ (tanθ θ) c) h(x)= 1+sinx x+cosx d) y=x 2 sinxtanx Example 3: Find the equation of the tangent line to the curve y= 1 at the point(0,1) sinx+cosx Section 3.3 Highly Suggested Homework Problems: 1, 5, 9, 13, 19, 21, 27, 35 (Do not simplify) 8

9 Section The Chain Rule The Chain Rule: If g is differentiable at x and f is differentiable at g(x), then the composite function m(x) = f(g(x))is differentiable at x and is given by In Leibniz notation, if y = f(u) and u = g(x) are both differentiable functions, then General Derivative Rules: If y=[ f(x)] n then If y=sin[ f(x)] then If y=e f(x) then If y=b f(x) then Example 1: Differentiate the following: a) f(x)=(4x 2 + 7x) 5 b) g(x)=6(x 1/2 3x) 4 c) y= 5 (t 2 + 3t+ 4) 4 d) F(x)=e x2 9

10 e) H(x)=3x 5 e x4 f) h(x)= 3 4 (3x+2) 5 g) s(t)= ( ) sint e t t 2 h) f(x)=(4x 2 + 5) 6 ( 3 (3x 4 5x+7) 4 ) i) G(x)=3 cos(x4 x) 10

11 Example 2: Find the value(s) of x where the tangent line is horizontal for f(x)= x 2 (2 3x) 3 Example 3: You are given the following information: g(4) = 8, g (4) = 3, f (4) = 7, f (8) = 6, f (3) = 5. If h(x)= f(g(x)), what is h (4)? Section 3.4 Highly Suggested Homework Problems: 1, 5, 9, 13, 17, 21, 25, 27, 31, 37, 41, 43, 45, 51, 53, 55, 69 (Do not simplify) 11

12 Section Derivatives of Logarithmic Functions Derivatives of Logs: If f(x)=lnx then If f(x)=log b x then If g(x)=ln( f(x)) then If g(x)=log b ( f(x)) then Example 1: Find the derivative of each of the following functions: a) f(x)=lnx+7 b) h(x)=4x 2 3 x+log 7 x 3 5 x c) y=ln(x 7 )+3(2) x ( 1+x ) d) y=log x ( ) 5 e) F(x)=ln x 6 12

13 ( ) x 1 f) y=log 7 x+2 g) y= 5ln((x2 + x) 5 ) x 3 h) f(x)=ln x Example 2: Find y and y for y= ln(3x) x 8 Section 3.7 Highly Suggested Homework Problems: 3, 5, 7, 9, 11, 17, 19, 21, 25, 27, 29 (Do not simplify) 13

14 Section Rates of Change in the Natural and Social Sciences Example 1: A particle moves according to a law of motion s= f(t)=0.01t t 3, t 0, where t is measured in seconds and s in feet. a) Find the velocity at time t. b) What is the velocity after 2 seconds? c) When is the particle at rest? d) When is the particle moving in the positive direction? e) Find the total distance traveled during the first 8 seconds. f) Draw a diagram to illustrate the motion of the particle. 14

15 g) Find the acceleration at time t and after 3 seconds. h) Graph the position, velocity, and acceleration functions for t 0. i) When is the particle speeding up? When is it slowing down? Example 2: In each of the following determine when the particle is speeding up and when the particle is slowing down. a) The graph of the position function of a particle is given where t is measured in seconds. s t s(t) b) The graph of the velocity function of a particle is given where t is measured in seconds. v t 1 2 v(t) 15

16 Example 3: If a tank holds 5000 gallons of wate, which drains from the bottom of the tank in 40 minutes, then Torricelli s Law gives the volume V of water remaining in the tank after t minutes as V = 5000( t)2, 0 t 40 Find the rate at which water is draining from the tank after (a) 5 minutes, (b) 10 minutes, and (c) 40 minutes. At what time is the water flowing out the fastest? The slowest? Example 4: The number of yeast cells in a laboratory culture is modeled by the function n= f(t)= a 1+be 0.2t where t is measured in hours. At time t = 0 the population is 405 cells and is increasing at a rate of 8.1 cells/hour. Find the values of a and b. According to this model, what happens to the yeast population in the long run? Section 3.8 Highly Suggested Homework Problems: 1, 3, 5, 7, 9, 13, 15, 23, 25 16

17 Section Linear Approximations and Differentials The approximation is called the linear approximation or tangent line approximation of f at a. The linear function whose graphs is this tangent line, that is, is called the linearization of f at a. Example 1: Find the linear approximation of the function f(x) = 3 1+x at a = 0 and use it to approximate the numbers and Example 2: Use a linear approximation to estimate(8.04) 4/3. Example 3: Find the linear approximation of the function f(x)=e x at a=0 and determine the values of x for which the linear approximation is accurate to within

18 Example 4: Suppose that we don t have a formula for g(x) but we know that g(2)= 4 and g (x)= x for all x. Use a linear approximation to estimate g(1.95) and g(2.05). If y = f(x), where f is a differentiable function, then the differential dx is an independent variable. The differential dy is then defined in terms of dx by the equation Example 5: Let s look at a graph to help us understand differentials. Example 6: Let y= x. Find the differential dy. Evaluate dy and y if x=1 and dx= x=1. Sketch a diagram showing the line segments with lengths dx, dy, and y. Section 3.9 Highly Suggested Homework Problems: 5, 7, 9, 11, 13, 15, 17, 23, 25, 35 18