12.3 Tangent to a Curve

Concept of tangents to curves and derivatives by examining lines tangent to a curve at a point, finding slopes, equations of tangent lines, and the first derivative of a function. Understand how to calculate the slope and equation of a tangent line for a given function at a specific point. Dive into the mathematical principles behind tangents and derivatives to enhance your problem-solving skills.

• Tangents
• Curves
• Derivatives
• Slope Calculation
• Equations

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1. 12.3 Tangent to a Curve

2. A line tangent to a curve is the line that passes through the point and that has a slope that is the same as the slope of the curve at that point.

3. Let A and B be two points near each other on the curve and let the coordinates of A be (x, f(x)). Suppose the x-coordinate of B differs from the x- coordinate of A by a small amount, h. Then the coordinates of B are (x + h, f(x + h)) What is the slope of the line? ( ) ( ) f x x ( ) ( ), + + f x h f x h f x = 0 h + x h h

4. Suppose A remains fixed while B moves a long the curve toward A. Then the value of h will become smaller, approaching zero. Thus, h can be considered as a variable that approaches zero as B approaches A. If B is made to coincide with A, then the secant line becomes tangent to the curve at point A. tangent secant ( ) ( ) f x + f x h lim h h 0

5. Find the slope of a line tangent to y = 2x2 3x + 1 ( ) ( ) lim h h ( ) ( ) 2 3 lim h h ( 2 2 3 lim h + f x h f x = 4 3 m x 0 ( ) 2 + + + + 2 1 2 3 1 x h x h x x 0 ) + + 1 2 + + 2 2 2 3 3 1 x xh h x h x x h 0 + 2 4 2 h 3 + + xh h h 2 2 2 2 4 2 h 3 2 x xh h h x = lim h lim h 0 0 ( ) = = + 4 3 x lim 4 h 2 3 x h 0

6. The first derivative (f(x)) of a function tells you the slope of all lines tangent to the function and is defined as: + ( ) ( ) f x f x h = '( ) f x lim h h 0

7. Find the equation of the line tangent to the graph at the indicated point: y = x2+ x 1 at (-4, 11) ( ) ( ) lim h h ( ) ( ) ( 1 lim h h 2 1 lim h h 2 2 lim h h ( ) ( ) ' 2 4 1 f x = + ( ) ' 7 f x = + = + + f x h f x y mx b ( 7 = 28 b = b = ) + 11 11 17 4 b 0 ) 2 + + + + 2 1 x h x h x x 0 = 7 17 y x + + + + + 2 2 2 1 x xh h x h x x 0 + + xh h h ( ) = + + = + lim 2 h 1 x h 2 1 x 0 0

8. Find the equation of the line tangent to the graph at the indicated point: 4 at (4, 1) y x ( ) ( ) lim h h 4 4 x h x h 0 h h 4 lim h x x h + 4 lim h x x h + 4 4 1 4 = ( ) ' 4 = = = f ( ) 4 2 16 + f x h f x = = + y 1 mx b ( ) 4 1 b = + b = 0 1 4 4 4 x x + h + 4 b ( ) x x h + lim h = lim 1 2 0 1 h h 1 4 ( ) = + 2 y x 0 4 x = ( ) 2 0

9. Find the derivative of the function f(x) = 8x2+ 2x 4 at x = 2. ( ) ( ) f x h f x h ( ) ( ) 8 2 4 x h x h h 2 2 8 16 8 2 x xh h x h 2 16 8 2 xh h h h ( ) 0 + lim h 0 ( ) 2 + + + + 2 8 2 4 x x lim h 0 + + + + 4 8 + 2 2 2 4 h x x lim h 0 ( ) ' 2 ( ) ' 2 ( ) = = + 16 2 34 2 f f + + lim h 0 ( ) + + = + lim 16 h 8 2 16 2 x h x

10. 12.3 b Derivatives Who uses this? Quantitative Analyst Financial Engineer Fund Accountant Engineering Researcher

11. The derivative of a function f(x) is another function f (x), that gives the slope of the tangent line to the function at any point. + ( ) ( ) f x f x h = '( ) f x lim h h 0

12. The process of finding the derivative is called differentiation. (this is one of the main parts of Calculus) Another common notation for dy '( ) is , ' dx This notation emphasizes that the derivative is a limit of slope, which is a change in y divided by a change in x. f x y

13. Ex 1 Find an expression for the slope of the tangent line to the graph of y = 2x2 3x + 4 at any point. = = ' 4 3 m y x Find the slopes of the tangent lines when x = -1 and x = 5. ( ) ( ) ' 1 4 1 3 7 y = = ( ) ' 5 ( ) 4 5 = 3 17 = y

14. Rules: 1. Constant: f(x) = constant f (x) = 0 if f(x) = xn(n is a rational #) then f (x) = nxn 1 2. Power: EX 2 f(x) = x3 ( ) ' f ( ) ' f 3 1 = = 3 3 x x x x 2

15. 3. Product of Constant & Power: f(x) = cxn then f (x) = cnxn - 1 EX 3: f(x) = 4x2 ( ) ' f x = 8 x 4. Sum & Difference: if f(x) = g(x) +/- h(x) then f (x)=g (x) +/- h (x) EX 4: f(x) = x3+ 5x2+ 6 ( ) ' 3 f x = + + = + 2 2 10 0 3 10 x x x x

16. 5. Product: ( dx ) means the derivative of u times v both d uv which are written in terms of x. d(uv) dx dx v u v v + = v du u dv dx What does this mean? Derivative of the first factor times the second factor plus the first factor times the derivative of the second factor. = + du ' ' u dx

17. = ' 2 2 u v = x = = + 2 3 7 u v x 2 Ex 5 ' x f(x) = (x2 + 3)(2x 7) ( ) ( 2 ' f x = ( ) ' 4 f x x = ( ) ' 6 f x x = ( ) ( ) 2 ) ( ) + + 2 x 3 x x 2 7 + + + 2 2 14 14 2 6 6 x x x 2 = = 2 + 2 ' 1 u = ' v = 1 u v x x 2 2 2 2 x x f(x)=(x + 1)(x2 2x) ( ) ( ) 1 ' f x = ( ) ' f x = ( ) ' f x = ( ) ( + ( x ) 1 ) + + x + x 2 x 2 2 x 2 x 2 2 2 2 2 2 x 3 x 2 x 2 x x

18. = u v v du dx u dv dx 6. Quotient: d 2 dx v u v ' ' u v u v = d 2 v x = 6: ( ) f x EX 1 x ( )( 1 ) ( )( ) ) 1 1 1 1 1 1 x x x x ( ) x = = = ' f ( ) ( ) 2 ( 2 2 x 1 x x

19. + 2 2 5 3 x x x = 7: ( ) f x EX 3 ( ) ( ) ) 5x x ( 2 x ( 5x x ) ( x 3 2 + 2 15 1 3 4 x ( ) x = ' f ) 2 3 = 3 5 v x x = ' + 2 2 3 u u x = ' 15 = 4 x 2 1 v x

20. ( ) + 2 3 7 3 1 2 x x = 8: ( ) f x EX 2 x x ( ) ( ) ( ) ( ) ) 2 3 + 2 6 x+ 7 1 x x 4 2 ( 3 2 x x + 3 2 x x 2 35 3 x x ( ) x = ' f 2 2 ( ) = = + + 2 3 2 v x x = ' 14 = ( ) ( x x 35 x 2 3 7 1 u u x x + )( ) 3 2 2 7 1 3 x x ' 6 2 v x = 4 2 ' 3 u x

21. 7. Power of a Polynomial (The Chain Rule): ( ) d u dx ( ) ( ) ( ) ( ) ' ' x n u u = n du dx = 1 n nu n = f x u 1 n f EX 9: f(x)=(3x 4)3 ( ) ' f x ( ) ' f x ( ( ) ( ) ) 2 = 3 3 4 3 x 2 = 9 3 4 x

22. ( ) ( f x )( ) 2 = + 3 4 3 7 2 5 x x x Ex 10 ( ) 2 = = + x 4 = ' 2 2 2 2 5 + 3 v v x ( 3 = 7 u x 9 x 7 ) 1 u x 4 3 ' 5 8 x ( ) ( ) ( ) ( ) ( x ) 2 x + + x + 4 3 3 4 = 2 5 2 3 7 16 2 5 x x x ' 9 7 f x

23. = 2 Ex 11 ( ) f x 1 x 1 2 ( ) ( f x ) = 2 1 x 1 2 1 2 ( ) ( 1 ( ) x ) = 2 ' 2 f x x x ( ) x = ' f 1 2 ( ) 2 1 x x ( ) x = ' f 2 1 x

24. = = = = ' ' 2 3 u v x ( 2 u v x ( 2 x ( ) f x = Ex 12 ) ( ) ) 3 2 3 + 2 1 x + 2 + 1 2 2 x x 1 x ( ) ( ) 3 2 x + x x + 2 2 2x 1 6 ) 1 2x ( x + ( ) x = ' f 6 2 1 ( ) ( )( f x = ) 3 + 2 2 1 x x )( ) ( ) x ( ) ( 1 3 4 ( ) ( x ) = + + 3 x + 2 2 2 ' 2 1 2 f x x x 3 2 6 + x + x ( ) x = + ' f ( ) ( ) 3 4 2 2 1 1 x x

25. WS 12.3