Derivatives and Rates of Change

Sec.1 Derivatives and Rates of Change A. Slope of Secant Functions rise Recall: Slope = m = = run Slope of the Secant Line to a Function: Examples: y y = y1. From this we are able to derive: x x x1 m y x 1 or y x 1 m 1. a.) Find the slope of the secant line to the function x x f x f x x x f between x 1 and x 9 1 1 b.) Find the equation of the secant line to the function x x f between x 1 and x 9 y x at the point.) Estimate the equation of the tangent line to the function 1,1 by calculating the equation of the secant line between x 1 and x 1. 1, between x 1 and x 1. 01 and between x 1 and x 1. 001. Desiré Taylor Math 141 1

A generalization of the previous example gives the definition: Slope = lim xa f x f x a a The slope of the tangent line can alternatively be calculated in the following way: In order to get the two points P 1 and P as close together as possible, we need for the space h 0. So, the slope between P 1 and P is: y m x f x h f x x h x f x h f x h but as the space h 0, we have m lim h0 f x h f x h B. Definition of Derivative The definition of a derivative (a.k.a. the slope of the tangent function) is given as: f x f x h f x lim or f a h0 h lim xa f x f x a a *Note: In this section we will use the DEFINITION OF THE DERIVATE to calculate all derivatives. (This means we will be doing it the long way!) Desiré Taylor Math 141

Examples: 1.) Find the equation of the tangent line to the function x x x 1 f where 5 x..) A person standing on top of a 00 ft tall building throws a ball into the air with a velocity of 96 ft/sec. The function s t 16t 96t 00 velocity of the ball at t = seconds gives the ball's height above ground, t seconds after it was thrown. Find the instantaneous Desiré Taylor Math 141

.) 4.) Desiré Taylor Math 141 4

5.) 6.) Desiré Taylor Math 141 5

Sec. The Derivative of a Function A. Definition of the Derivative For a function f x the derivative is f x lim h0 f x h f x h Examples: Using the definition of the derivative, find the derivative of the following functions: 1.) f x x x.) g x x 4 Desiré Taylor Math 141 1

.) k x 1 x 1 4.) Consider the graph for the function f x Estimate the following: a.) c.) f A b.) f B f C d.) f D e.) f E f.) f F Desiré Taylor Math 141

B. Differentiability DEFN: We say that x x f is differentiable at a if f a exists f is differentiable on a, b if it is differentiable on every point in a, b Theorem: If f is differentiable at a, then f is continuous at a Non differentiable functions: i. Any function with a "corner" or cusp ii. Any function with a discontinuity iii. Any function with a vertical tangent Desiré Taylor Math 141

C. Notation Function y y f x f x Derivative (Leibniz notation) F x f x D. Higher Derivatives (i.e. Finding the "derivative of a derivative" or taking the derivative multiple times.) d d y y f x lim h0 f x h f x h Example: f ) 1.) For the function f x x x, find the second derivative. (Recall from previous that x x Desiré Taylor Math 141 4

More Examples: 1.) Identify the graphs A (blue), B( red) and C (green) as the graphs of a function and its derivatives. a.) The graph of the function is: b.) The graph of the function's first derivative is: c.) The graph of the function's second derivative is:.).) Desiré Taylor Math 141 5

Sec. Basic Differentiation A. Properties and Formulas (The short way Yeah!) 1. Basic Functions Function Derivative f x c (Constant) f x 0 f x x f x 1 f x cx f x c n f x x n1 f x nx n f x cx n1 f x cnx f f f x c gx f x c gx x gx hx f x gx hx x gx hx f x gx hx Note: For the function f x gxhx g For the function x x hx we CAN NOT say that f x gx hx f g we CAN NOT say that x f x hx. Trigonometric Functions Function Derivative f f f f f f x sinx f x cos x x cosx f x sin x x tanx f x sec x x cscx f x csc xcotx x secx f x sec xtanx x cotx f x csc x Desiré Taylor Math 141 1

Examples: Find and LABEL the derivatives of each of the following functions. 1.) f x 8.) gx 8x.) kx x 4.) hx 5x r 4 5.) v r 6.) qy y a x 4x 7.) mx x 8.) 1 t 9.) vt 10.) 1 x x 4x x 11.) px 1.) x Desiré Taylor Math 141

More Examples 1.) f x x 4x x 4 Find f 1 4t t 5 14.) gt Find gt t 5 4 15.) x x x x x 6 h Find the first 5 derivatives of the function. Desiré Taylor Math 141

B. Normal and Tangent Lines to a Function Normal Line: A line that is perpendicular to the tangent line. Examples: 1.) Find the tangent and the normal lines to the function f x 4cosx at x.) Find the horizontal tangent lines (lines with slope = 0) to the function f x x x 10x Desiré Taylor Math 141 4

C. Applications to Position, Velocity and Acceleration If the motion/position function of a particle is known, we can find the velocity and acceleration functions in the following way. If the position of a particle is given by f x, then the velocity of the particle is given by f x If the velocity of a particle is given by x (We can also say that If the position of a particle is given by f given by, the second derivative of the motion function.) f x g, then the acceleration of the particle is given by g x x, then the acceleration of the particle is Alternative notation: Position of a particle s t Velocity of a particle v t Acceleration of a particle a t Then vt st And at vt st Example: 1.) A particle's position is described by the function st t 144t a. Find the velocity function.. ( t is measures in seconds and s t in feet.) b. Find the acceleration function. c. Find the acceleration after 9 seconds. d. Find the acceleration when the velocity is 0. Desiré Taylor Math 141 5

.) A particle's position is described by the function f t t 9t 15t 10 feet.) a. Find the velocity function.. ( t is measures in seconds and s t in b. What is the velocity after seconds? c. When is the particle at rest? d. When is the particle moving in a positive direction? e. When is the particle slowing down? f. Find the total distance traveled during the first 8 seconds. Desiré Taylor Math 141 6

A r r.) The area of a disc with radius r is when r 5.. Find the rate of change of the area of the disc with respect to its radius More Examples: 1.).) a.). b.) c.) Desiré Taylor Math 141 7

4 a.) b.) 5 a.) b.) Desiré Taylor Math 141 8

Sec.4 Product and Quotient Rules A. Product Rule f x gxhx then f x gx hx gxhx Alternative Notation: d g h dg dg h g In Plain English: The derivative of the product of two functions (which we will call the first function and the second function) is equal to the derivative of the first, times the second, plus the first, times derivative of the second. Examples 4 1.) f x x 8x 5x 17 We see that f x consists of the product of two smaller functions, in this case x 8x the second. So, the derivative then is: the first and5x 4 17 f x 6 x 8 x 5x 4 17 + x 8x x 0x Note: Derivative of the first You should leave the answer in this form unless we are asked to clean up Again, do not forget to label your derivative.) gx x sinx We see that gx consists of the product of two smaller functions, in this case x the first and x So, the derivative then is: gx 1 sinx x cosx sinx x cosx More Examples: Find and LABEL the derivatives of each of the following functions. 4 1.) f x x x 5x 1 The second The first Derivative of the second sin the second..) f x x cosx.) f x sinxcos x Desiré Taylor Math 141 1

B. Quotient Rule g x f x then f x hx g x hx gx hx hx Book Notation: g d h dg h h g dg In Plain English: The derivative of the quotient of two functions (which we will call the top function and the bottom function) is equal to the derivative of the top, times the bottom, minus the top, times derivative of the bottom, all over the bottom squared. Examples 1.) x 8x f x 4 5 x 17 We see that f x consists of the quotient of two smaller functions, in this case x 8x the bottom. So, the derivative then is: Derivative of the top The bottom The top the top and5x 4 17 Derivative of the bottom f x 6 x 8 x 5x 17 - x 8x x 0x 4 5 x 17 The bottom squared Note: You should leave the answer in this form unless we are asked to clean up Again, do not forget to label your derivative.) gx We see that x x gx consists of the product of two smaller functions, in this case x the top and x sin So, the derivative then is: x sinx x cosx sin x x 1 sin x cos sin g x x sin the bottom. Desiré Taylor Math 141

More Examples: Find and LABEL the derivatives of each of the following functions. 4 1.) x 8x f x x 1.) gx sin x cos x sin x.) lx x x x 4.) Desiré Taylor Math 141

5.) a. b. c. d. e. fg 7 f. g f 0 6.) f is the bottom function g is the top function 7.) Desiré Taylor Math 141 4

Sec.5 Chain Rule A. The Chain Rule Alternative Notation: f h hgx hgx gx x h gx hgx then f x hg x gx g x hg d dh dg dg In Plain English: First, identify which function is on the outside and which is on the inside. (For the composition f gx f gx we say that f is on the outside and g is on the inside.) The derivative of this composition is equal to the derivative of the outside (leave the inside alone) times the derivative of the inside. Examples f x sin(5x 5 1.) ) First, let us identify which is the outside and which is on the inside : Here sin is the outside (i.e. sin of something ) and 5 5x is the inside. Derivative of the outside is Derivative of the inside is 5 cos, and if we the leave the inside alone, this will be cos5 x 4 5x 5 4 f x cos 5 x x 5x Derivative of the outside (leave the inside alone) Derivative of the inside Note: This style of answer should only be cleaned up if you are given specific instructions to so (or if you have to compare it to a list of multiple choice answers)! Again, do not forget to label your derivative Desiré Taylor Math 141 1

More Examples.) gx sin x sin x We see that gx consists of So, the derivative of the outside is g x sin x cos x x and derivative of the inside is cos x as the outside andsin as the inside. Examples: Find and LABEL the derivatives of each of the following functions. 1.) f x tan6 x.) kx tansin x.) lx x 1 4.) f x x x 8 5.) rs s 4 1 s Desiré Taylor Math 141

B. Combinations of Product, Quotient and Chain Rule In many problems we need to use a combination of the Product, Quotient and Chain Rule to find a derivative. Here we will work through lots of examples. Examples: Find and LABEL the derivatives of each of the following functions. Do not clean up unless otherwise indicated. 1.) gx sinx 8 x 1.) f x sin x 4.) gx x cos x 4.) r cos 4 Desiré Taylor Math 141

5.) ax x x 1 6.) f x 5x x 5x 7.) 8.) 9.) Desiré Taylor Math 141 4

10.) 11.) Desiré Taylor Math 141 5

Sec.6 Differentiation of Implicit Functions Recall from the previous week, that when we take the derivative of function in terms of x (i.e. the only variable in the function is x) y f x then y f x where f x is a Example: If y x x then y x So, in other words, to take a derivative this way, we have to have the equation solved for y. Example: If x x x y then y x 1. Here we first have to solve for y. So y x x x y x x Example: If yx x xy y. Again, we have to solve for y in order to take a derivative in the way that we have learnt in the preceding chapters. However, (as you can see in this case) it is not always easy/possible to do so. A. Implicit Differentiation *HENCE: Implicit Differentiation!* Implicit Differentiation: Differentiation of a function where one variable (typically y) is not explicitly expressed a s a function of another variable (typically x). Here s how it works: It is important to pay attention to the notation. If we are given an equation in terms of x and y, and asked to find y or, we need to see that we are finding the derivative of y, with respect to x. We will treat both x and y like a variable, and take derivatives of each, but; When we take a derivative of a term containing x we will proceed as usual When we take a derivative of a term containing y we will proceed as usual AND then also multiply the derivative of that term by (or y ). We will use product, quotient and chain rules as needed. After differentiating, solve for (i.e. isolate) y or, Example: Find for x x y. x d 1 x y Since we are trying to find, isolate in our equation: 1 x y 1 x y Desiré Taylor Math 141 1

Example: Find for x d x d8 y d x 4x y 8y. 4 1 1x y 8 Since we are trying to find, isolate in our equation: 11x y 8 11x y 8 11x y 8 Example: Find y for x x y y. (Notice that in this problem we have will have to use the product rule.) x y d y 1 x y x y y 9y y d x y x y y 9 y y 1 x y yx y 9 1 y x y - a product of x and y. Here we 1 x y x y 9y y Examples: Find for the following: 1.) x y 11 0 Desiré Taylor Math 141

5y.) y x x 4 x 1.) Find y for 5 x y 0 y and evaluate at, 1 4.) Find da for A r dt Desiré Taylor Math 141

5.) Find dv for dt 1 V r h 6.) 7.) Desiré Taylor Math 141 4

Sec.7 Related Rates Before getting started with Related Rates, let us re-visit the following items first: Notation, Implicit Differentiation and Geometric Formulas A. Notation f x y Although all of the above notations are equivalent, we will use Leibniz's notation in this section, because it is more descriptive than the other forms. Leibniz's notation tells us specifically what we are taking a derivative of (in this case the function y) and what we are taking the derivative with respect to (w.r.t.) i.e. what is the variable in the function (in this case x.) B. Implicit Differentiation Again, we will have to pay close attention to notation here. In equations with multiple variables, we will be asked to find derivatives of specific parts of the equations with respect to specific variables (that may or may not be part of the equation!). For Example: Consider the equation for the circle: r x y dr We would like to find. This means, we are trying to find the derivative of r with respect to t. I.e. take a derivative of dt each term with respect to t. (If a term is/contains a t, just take a derivative as usual. If a term contains a variable other than t, follow the usual rules for implicit differentiation.) dr dt dt So, r x y dt dr dt dt r x y dt We would like to find. dt dr dt dt So, r x y dt dt dr dt x r y dt Desiré Taylor Math 141 1

C. Geometric Formulas * You are responsible for knowing these formulas for all tests and the final exam* Shape Square -Dimentional Shapes Perimeter/Circumference and Area P 4S A S -Dimentional Shapes Shape Sphere SA 4R 4 V R Rectangle P H L Cylinder SA RH R V R H Trapezoid A H B b Cone V R H Parallelogram A BH Cube SA 6S V S Circle C R A R Rectangular Parallelepiped SA HB BW HW V BHW Triangle BH A Right Triangle c a b Desiré Taylor Math 141

D. Related Rate Problems dz 1. If z x y, 9, and 5, Find when x and y 5 dt dt dt. Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 1.5 m/s, how fast is the area of the spill increasing when the radius is 19 m? Hint: The word rate = derivative. Pay attention to units to find out which rate is given asked for. d length dr (Example, rate of 1.5 m/s meters per second = unit of length per unit of time = ) d time d t Desiré Taylor Math 141

. A fireman is on top of a 75 foot ladder that is leaning against a burning building. If someone has tided Sparky (the fire dog) to the bottom of the ladder and Sparky takes off after a cat at a rate of 6 ft/sec, then what is the rate of change of the fireman on top of the ladder when the ladder is 5 feet off the ground? 4. A street light is mounted at the top of a 11 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 8 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 50 ft from the base of the pole? Desiré Taylor Math 141 4

cm 5. If a snowball melts so that its surface area decreases at a rate of.01 min decreases when the diameter is 8cm., find the rate at which the diameter 6. At noon, ship A is 0 miles due west of ship B. Ship A is sailing west at 5 mph and ship B is sailing north at 18 mph. How fast (in mph) is the distance between the ships changing at 5 PM? Desiré Taylor Math 141 5

7. Water pours into in inverted cone at a rate of m /min. It the cone has a radius of m and a height of 4 m, find the rate at which the water level is rising when the water is m deep. Desiré Taylor Math 141 6

8.) A boat is pulled into a dock by means of a rope attached to a pulley on the dock. The rope is attached to the front of the boat, which is 7 feet below the level of the pulley. If the rope is pulled through the pulley at a rate of 0 ft/min, at what rate will the boat be approaching the dock when 10 ft of rope is out? * Examples on this topic available at Justmathtutoring.com > Free Calculus Videos > Related Rates - * Desiré Taylor Math 141 7

Sec.8 Linear Approximation and Differentials A. Differentials Slope = y x Slope of the tangent line = y x Also, Example: Find the differential of y given that: 1.) y 5x f x f x.) x 4x 5 y 5sin x x Note: x and y x y x x x f x x f x Desiré Taylor Math 141 1

Example: y x 1a.) Calculate y for x to x. 01 b.) Calculate for x to x. 01.) 4 V r (volume of a sphere) a.) Use differentials to approximate the change in volume when going from r ft to r. 8 ft dv (Find dv : Start by finding ) dr b.) Find the actual change in volume when going from r ft to r. 8 ft (Find V ) Desiré Taylor Math 141

B. Linearization Definition: The linearization of a function Slope of the tangent line = m T f L a Point Slope: y y m x L o T x 0 y f a f a x a y f a f a x a x f a f a x a f x at a fixed point a is given by the formula x f a f a x a Examples: 1.) Find the linear approximation of f x cos5 x at a.) Use linearization techniques to approximate 16. 1 Desiré Taylor Math 141

More Examples:.) 4.) Desiré Taylor Math 141 4

5.) The edge of a cube was found to be 60 cm with a possible error of 0.5 cm. Use differentials to estimate: (a) the maximum possible error in the volume of the cube (b) the relative error in the volume of the cube (c) the percentage error in the volume of the cube 6.) A 1 foot ladder is leaning against a wall. If the top slips down the wall at a rate of 4 ft/s, how fast will the foot of the ladder be moving away from the wall when the top is 8 feet above the ground? Desiré Taylor Math 141 5