Section 2.1, Section 3.1 Rate of change, Tangents and Derivatives at a point

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1 Section 2.1, Section 3.1 Rate of change, Tangents and Derivatives at a point Line through P and Q approaches to the tangent line at P as Q approaches P. That is as a + h a = h gets smaller. Slope of the line passing through P and Q = y x = f(a + h) f(a) a + h a = f(a + h) f(a) h = rate of change of y=f(x) with respect to x Slope of the tangent line passing through P = lim h 0 f(a + h) f(a) h = instantenous rate of change of y=f(x) with respect to x at x=a Definition 1 The derivative of a function f at a number a, denoted by f (a) is f (a) = lim h 0 f(a + h) f(a) h if the limit exists. Another way of expressing f (a) is f (a) = lim x a f(x) f(a) x a Example 2 Find the equation of the tangent line to the parabola y = 4x x 2 at the point (1, 3). 1

2 Derivative of a function f(x) at a point x=a = slope of the tangent line to the curve defined by y=f(x) at x=a = rate of change in y-values with respect to x-values f (a) = lim h 0 f(a + h) f(a) h where y = f(x) f(a), x = x a Derivatives calculate the changes occured = lim x a f(x) f(a) x a = lim x 0 y x Example 3 If a rock is thrown upward with a velocity of 10m/s, its height in meters after t seconds is given by H(t) = 10t 1.86t 2 (a) Find the velocity of the rock after one second. (b) Find the velocity of the rock when t=a (c) When will the rock hit the surface? (d) With what velocity will the rock hit the surface? 2

3 Example 4 The cost of producing x ounces of gold from a new gold mine is C=f(x) dollars. (a) What is the meaning of derivative of f(x)? What are its units? (b) What does the statement f (800) = 17 means? Recall that derivative of f(x) gives the slope of the tangent line to the curve y=f(x). Large Derivative = curve is relatively steep, or y coordinates change rapidly. Small Derivative = curve is relatively flat, or y coordinates change slowly Negative Derivative =curve is decreasing Positive Derivative = curve is increasing Example 5 x=1/2. 1. Compare the derivatives of f(x) = x 2 and g(x) = x 3 at x=2 and 2. Compare the derivatives of f(x) = 1/x and g(x) = 1/x 2 at x=1. 3

4 Example 6 Compare values 0, f ( 2), f (0), f (2), f (4) by looking at the graph of f(x) given below Example 7 Determine whether f (0) exists for f(x) = { x sin ( 1 x ) if x 0 0 if x = 0 4

5 Section 3.2 The Derivative as a Function Definition 8 Derivative of a function f is defined as f (0) = lim h 0 f(x + h) f(x) h Domain of f = {x Domain of f f (x) exists } Notations: f (x) = y = dy dx = df dx = df(x) dx = Df(x), f (a) = dy dx x=a Definition 9 A function is differentiable at x=a if f (a) exists. A function is differentiable on an open interval (a,b) if it is differentiable at every number in the interval. Classical example: Where the function f(x) = x is differentiable? 5

6 Theorem 10 if f is differentiable at x=a, then f is continuous at x=a Note 11 Converse statement of the theorem is not true. That is if f is a continuous fucntion it does not imply f is differentiable. Example 12 f(x) = x is continuous at x=0, but not differentiable at x=0. Example 13 Is the function f continuous and differentiable at x=2. { 1 2 f(x) = x + 2 if x > 2 x + 5 if x 2 Note 14 If a function has a graph with a cusp (sharp point, corner) then that function if not differentiable at that point. x 2 if x < 0 Example 15 Draw the graph of f(x) = 1 x if 0 x 2. Then determine the (x 3) 1 3 if x > 2 x values for which the function is not continuous and is not differentiable. 6

7 Vertical tangent: This occurs at a point x=a, where f is continuous at x=a and lim x f (x) = Example 16 Function f defined in the above example has a vertical tangent at x=3. 7

8 Section 3.3 Differentiation Rules 1. Derivative of a constant function: If f(x)=c, where c is a constant then f (x) = 0. Example 17 f(x) = 100 f (x) = 0, g(x) = e 1 0 g (x) = 0 2. Power Rule: If f(x) = x n, where n is a real number, then f (x) = nx n 1. Example 18 (a) f(x) = x 100 f (x) = (b) g(x) = x 25 g (x) = (c) h(x) = 5 x 2 h (x) = 3. Sum Rule and Difference Rule: If f and g are both differentiable then (f(x) ± g(x)) = f (x) ± g (x) Example 19 f(x) = x x 78 x 25 f (x) = 4. The Constant Multiple Rule: If f(x)=cg(x) and g(x) is differentiable, c is a constant then f (x) = cg (x) Example 20 f(x) = 10x x 4 + 3x f (x) = 5. Product Rule: If f and g are both differentiable then [f(x)g(x)] = f (x)g(x) + f(x)g (x) Example 21 f(x) = (2x + 3)(1 x) f (x) = 6. Quotient Rule: If f and g are both differentiable then [ f(x) g(x) ] = f (x)g(x) f(x)g (x) [g(x)] 2 Example 22 f(x) = 2x5 +6x 3 +4 x 4 2x f (x) = 8

9 Example 23 Find equations of the tangent lines to the curve y = x 1 that are parallel x+1 to the line x-2y=2. Higher Derivatives: If f is differentiable function with differentiable derivative then we can define the second derivative of f denoted by f as the derivative of f. f (x) = [f (x)] another notation is d dy dx dx = d2 y dx 2 If f is differentiable then we can define the third derivative of f as In general f (3) (x) = [f (x)] or denote as d d 2 y dx dx = d3 y 2 dx 3 f (n) (x) = [f (n 1) ] or denote as d d n 1 y dx dx = dn y n 1 dx n Example 24 The equation y + y 2y = x 2 is called a differential equation. constants A,B,C such that the function y = Ax 2 + Bx + C satisfies the equation. Find 9

10 Section 3.5 Derivatives of Trigonometric Functions: Note 25 Derivatives of other trigonometric functions can be found using derivatives of sin (x) and cos (x). Example Find d tan (x) dx 2. Find d cot (x) dx 3. Find d sec (x) dx sin (x) = cos (x) and cos (x) = sin (x) Example 27 A mass on a spring vibrates horizontally on a smooth level surface, Its equation of motion is given by x(t) = 8 sin (t) where t is in seconds and x in centimeters. (a) Find the velocity and acceleration at time t. (b) Find the position, velocity and acceleration of mass at time t = 2π. In what direction 3 is it moving at that time? 10