1 The Derivative and Differrentiability

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1 1 The Derivative and Differrentiability 1.1 Derivatives and rate of change Exercise 1 Find the equation of the tangent line to f (x) = x 2 at the point (1, 1). Exercise 2 Suppose that a ball is dropped from the upper observation deck of a tall tower, 450 m above the ground. a) What is the velocity of the ball after 5 seconds? b) How fast is the ball traveling when it hits the ground? Definition 1 Let f : A R R and a A an interior point of A. The derivative of f at a is the limit if it exists and it is finite. f f (x) f (a) (a), (1) x a x a Remark 2 Writing x = a + h (thus h 0), the above definition is the same as the following f f (a + h) f (a) (a), (2) Exercise 3 Find the derivative of the function at the f (x) = x 2 8x + 9 at a. Definition 3 The tangent line to the graph of f at the point (a, f (a)) is the line through this point and with slope f (a), that is, the line with equation y f (a) = f (a) (x a) (if f (a) exists and is finite). Exercise 4 Find an equation of the tangent line to the parabola y = x 2 8x + 9 at the point (3, 6). Remark 4 By analogy with average speed and instantaneous speed, for a given function y = f (x) we introduce the rate of change of f over the interval [x 1, x 2 ] as change in f change in x = y x = f (x 2) f (x 1 ), x 2 x 1 and the instantaneous rate of change at x = x 1 as the limit if this limit exists. y lim x 0 x f (x 2 ) f (x 1 ) = f (x 1 ), x 2 x 1 x 2 x 1 Remark 5 The instantaneous rate of change of f at x = x 1 is just the derivative f (x 1 ). 1.2 The derivative function and differentiability The derivative of a given function f is the function f defined by defined for all values for which this limit exists. f f (x + h) f (x) (x), Example 1 Find the derivative of the function f (x) = x and indicate its domain. To find the derivative of the function f (x) = x at a > 0, we compute f (x) f (a) x a lim x a x a x a x a x a x a (x a) ( x + a) x a 1 = 1 x + a 2 a, which exists and it is finite for any a > 0. This shows that the derivative of f (x) = x is f (x) = 1 2 x, x > 0. 1

2 10 f x 5 f x 0 5 Figure 1: The graphs of f(x) and f (x) in Exercise 5. Exercise 5 If f (x) = x 3 x, find a fomula for f (x). Draw a rough sketch of the graph of f directly from that of f. Exercise 6 Find f is f (x) = 1 x 2 + x. Notation 6 For a given function y = f (x) there are several possible notation for its derivative, as follows f (x) = y = df dx = dy dx = d dx f (x). The notation dy dy of the derivative is called Leibniz notation (it is suggestive for the fact that dx dx y x 0 x ). When we want to use Leibniz notation in order to indicate the value of the derivative at a certain point a, we write dy dx. x=a Definition 7 Let f : A R R and a A an interior point of A. We say that f is differentiable at a if the derivative f (a) exists. We say that f is differentiable on set if it is differentiable at every point of that set. Example 2 Where is the function f (x) = x differentiable? If x > 0, then f f (x + h) f (x) x + h x (why?) x + h x (x) = 1, so f is differentiable on (0, ). Similarly, f (x) = 1 for x < 0, and f is differentiable on (, 0). Since the limit f (0 + h) f (0) lim does not exist (limit is 1 from the left and +1 from the right), f is not differentiable at 0. The function f is differentiable on (, 0) (0, ). Theorem 8 If f is differentiable at a, then f is continuous at a. Proof. ( ) f (x) f (a) lim f (x) = f (a) + lim (x a) = f (a) + f (a) 0 = f (a). x a x x 0 x a Remark 9 The converse of the theorem is not true. To see this, consider the previous example: f (x) = x is continuous at 0 but it is not differentiable at 0. 2

3 Figure 2: Graphs of discontinuous functions at a: a) sharp corner at a b) discontinuous at a vertical tangent at a. Remark 10 Visually (by looking at the graph), we can see that a function is not differentiable at a point if it discontinuous at that point, if it has a sharp corner at that point, or if it has a vertical tangent at that point Definition 11 If a function f is differentiable, and it derivative f is again differentiable, we say that f is twice differentiable. We call (f ) the second derivative of f, and write f d 2 f d 2 y (x) or dx 2 or dx 2 = d ( ) dy. dx dx Exercise 7 If f (x) = x 3 x, find and interpret f (x). Definition 12 If f exist and it is again differentiable, we call the derivative (f (x)) = f (x) the third derivative of f. Inductively, f (n) represents the n th derivative of f (obtain by differentiating n times the given function f). Exercise 8 If f (x) = x 3 x, find f and f (4). 1.3 Basic differentiation formulas Example 3 Show that the derivative of a constant function is 0, that is (c) = dc dx = 0 (c R a constant). Example 4 Show that dx n dx = nxn 1, for any n R. When n in a positive integer, we have dx n dx (x + h) n x n ( ) x n + nx n 1 h + n(n 1) 2 x n 2 h nxh n 1 + h n x n h 0 h ( nx n 1 + h 0 = nx n 1. ) n (n 1) x n 2 h nxh n 2 + h n 1 2 Exercise 9 Find the indicated derivatives: a) If f (x) = x 6, then f (x) = b) If y = x 1000, then y = c) If y = t 4, then dy dt = Exercise 10 Find the derivatives of the given functions: a) f (x) = 1 x b) f (x) = x c) f (x) = 1 x 2 d) f (x) = 3 x 2 d) d dr ( r 3 ) = 3

4 Proposition 13 (Constant multiple, sum and difference rule) If f and g are differentiable functions, then: i) cf is also differentiable, and we have ii) f ± g is also differentiable, and we have (cf) (x) = cf (x), c R (f ± g) (x) = f (x) ± g (x) Exercise( 11 Find the indicated derivatives: a) ) d dx 3x 4 b) d dx ( x) c) ( d dx x x 5 4x x 3 6x + 5 ) Exercise 12 Find the points on the curve y = x 4 6x where the tangent line is horizontal. Example 5 Show that We have d dx (sin x) = cos x and d (cos x) = sin x. (3) dx d sin (x + h) sin x (sin x) dx (cosh 1) sin x = sin x 0 + cos x 1 = cos x. h 0 sin x cosh + cos x sinh sin x h + cos x sinh h Exercise 13 Find the indicated derivatives a) y = 3 sin θ + 4 sin θ b) 5 th derivative of sin x c) 27 th derivative of cos x. Exercise 14 (An application of derivatives in Physics) The position of a particle is given by s = f (t) = t 3 6t 2 + 9t (time t measured in seconds and distance s measured in meters). a) Find the velocity at time t b) What is the velocity after 2 seconds? After 4 seconds? c) When is the particle at rest? d) When is the particle moving forward? e) Draw the position function of the particle. f) Find the total distance traveled by the particle during the first five seconds. g) Find the acceleration of the particle at time t, and at t = 4. h) Graph the position, velocity, and acceleration functions on the interval [0, 5]. 1.4 The product and quotient rule Proposition 14 (The product and quotient rule) If f and g are differentiable functions, then: i) f g is also differentiable, and we have (f g) (x) = f (x) g (x) + f (x) g (x) ii) if g (x) 0, then f g is also differentiable, and we have ( ) f (x) = f (x) g (x) f (x) g (x) g g 2. (x) 4

5 Exercise 15 Find the derivatives of the given functions a) f (x) = x 2 sin x b) f (t) = t (a + bt) c) f (x) = x2 + x 2 x 3 q + 6 Exercise 16 Find h (3), if h (x) = x, g (3) = 5, and g (3) = 2. Exercise 17 Find the equation of the tangent line to the curve y = Exercise 18 Find the derivatives of the given functions a) f (x) = tan x b) f (x) = cot x c) f (x) = sec x d) f (x) = csc x. Exercise 19 Find the derivatives of f (x) = 1.5 The chain rule d) f (x) = 3x2 + 2 x x ( x at the point 1, 1 ). 1 + x2 2 sec x 1+tan x. Where does the graph of f have a horizontal tangent? Theorem 15 If f and g are differentiable functions (g differentiable at x, and f differentiable at g (x)), then f g is also differentiable, and (f g) (x) = f (g (x)) g (x). Exercise 20 Find the derivatives of the given functions. a) f (x) = x b) f (x) = sin ( x 2) c) f (x) = sin 2 x d) f (x) = ( x 3 1 ) 100 ( ) 9 1 t 2 e) f (x) = 3 x f) g (t) = 2 +x+1 2t+1 g) f (x) = (2x + 1) 5 ( x 3 x + 1 ) 4 h) f (x) = sin (cos (tan x)) i) f (x) = sec (x 3 ) 1.6 Implicit differentiation Idea: sometimes, the function we are interested is given by an implicit rather than an explicit equation. To find the derivative of the function, we simply differentiate the given equation (and solve) in order to obtain the derivative. This technique is called implicit differentiation. Exercise 21 If x 2 + y 2 = 25, find dy dx : a) directly (solve for y, then differentiate) b) using implicit differentiation. Which method seems to be easier? Find the equation of the tangent line to the circle x 2 + y 2 = 25 at the point (3, 4). Exercise 22 Consider the curve x 3 + y 3 = 6xy. a) Find y b) Find the equation of the tangent line to the curve at the point (3, 3) c) At what point in the first quadrant is the tangent line horizontal? Exercise 23 Find y if sin (x + y) = y 2 cos x. Exercise 24 Find y if x 4 + y 4 = Applications (related rates) Exercise 25 Air is being pumped into a spherical balloon so that its volume increases at a rate of 100 cm 3 /sec. How fast is the radius of the balloon increasing when the diameter is 50 cm? Exercise 26 A 10 ft long ladder rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall? Exercise 27 A water tank has the shape of an inverted circular cone with base radius 2 m and height 4 m (draw a picture!). If water is being pumped into the tank at a rate of 2 m 3 /min, find the rate at which the water level is rising when the water is 3 m deep. Exercise 28 Car A is traveling west at 50 mi/h and car B is traveling north at 60 mi/h. Both are headed for the intersection of the two roads. At what rate are the cars approaching each other when car A is 0.3 mi and car B is 0.4 mi from the intersection? Exercise 29 A man walks along a straight path at a speed of 4 ft/s. A searchlight is located on the ground 20 ft from the path and is kept focused on the man. At what rate is the searchlight rotating when the man is 15 ft from the point on the path closest to the searchlight? 5

6 1.8 Linear approximation and differentials If a function f is differentiable at a, then the tangent line is a good approximation of f near a (see figure below). Recalling that the equation of the tangent line is y f (a) = f (a) (x a) (or y = f (a) + f (a) (x a)), this indicates that we have the following approximation near a f (x) f (a) + f (a) (x a) = L (x), called the linear approximation or tangent line approximation (L (x) is called the linearization of f at a). Figure 3: Tangent line approximation (blue) of the function (black). Exercise 30 Find the linearization of the function f (x) = x + 3 at a = 1 and use it to approximate the numbers 3.98and Are these approximations overestimates or underestimates? We have seen that if f is differentiable at x, then so for small x we have We define the differential of f at x by f (x) x 0 f (x + x) f (x), x f = f (x + x) f (x) f (x) x. df (x) = f (x) dx, and interpret it as the change in f for a small change dx in the variable x (the formula actually gives the change in the tangent line for the corresponding change dx in x). Example 6 The radius of a sphere was measured and found to be 21 cm with a possible error in measurement of at most 0.05 cm. What is the maximum error in using this value of the radius to compute the volume of the sphere? The volume of a sphere is V = 4 3 πr3. In terms of differentials, this means dv = 4πr 2 dr. Since the error in measurement of the radius dr = 21 cm is small, we obtain that the error in measurement of the volume can be approximated by the differential dv = 4π (21) cm 3. 6