Aim: Differentials Course: Calculus Do Now: Aim: Differential? Isn’t that part of a car’s drive transmission? Find the equation of the tangent line for f(x) = 1 + sinx at (0, 1).

Aim: Differentials Course: Calculus Linear Approximations graph of function is approximated by a straight line. y = x 2 = 2x - 1 y = x 2 = 2x - 1

Aim: Differentials Course: Calculus Linear Approximations c c, f(c) y 2 – y 1 = m(x 2 – x 1 ) - point slope By restricting values of x to be close to c, the values of y of the tangent line can be used as approximations of the values of f. y– f(c) =f’(c)(x – c) y = f(c) + f’(c)(x – c) x x (x, y) Can the graph of a function be approximated by a straight line? as x  c, the limit of s(x) or y is f(c) Equation of tangent line approximation f s equation of tangent line

Aim: Differentials Course: Calculus Model Problem Find the tangent line approximation of at the point (0, 1). 1 st derivative of f y = f(c) + f’(c)(x – c) y= 1 +cos 0(x – 0) y= 1 +1x1x The closer x is to 0, the better the approximation. = 1 + x Equation of tangent line approximation

Aim: Differentials Course: Calculus Differential derivative of y with respect to x When we talk only of dy or dx we talk differentials As Δx gets smaller and smaller, before it reaches 0,  approximates ΔyΔy actual change dy approximation of Δy also the ratio dy  dx is the slope of the tangent line

Aim: Differentials Course: Calculus Differential Approximations c c, f(c) c + Δx ΔxΔx f(c)f(c) ΔyΔy f’(c)Δx (c + Δx, f(c + Δx)f(c + Δx) When Δx is small, then Δy is also small and Δy = f(c + Δx) – f(c) and is approximated by f’(c)Δx. dy = dx the ratio dy  dx is the slope of the tangent line

Aim: Differentials Course: Calculus Differential When Δx is small, then Δy is also small and Δy = f(c + Δx) – f(c) and is approximated by f’(c)Δx. Δy = f(c + Δx) – f(c)actual change in y  f’(c)Δx approximate change in y Δy  dyΔy  f’(c)Δx Let y = f(x) represent a function that is differentiable in an open interval containing x. The differential of x (denoted by dx) is any nonzero real number. The differential of y (denoted by dy) is dy = f’(x)dx Definition

Aim: Differentials Course: Calculus Differential Δx is an arbitrary increment of the independent variable x. Definition dx is called the differential of the independent variable x, dx is equal to Δx. Δy is the actual change in the variable y as x changes from x to x + Δx; that is, Δy = f(x + Δx) – f(x) dy, called the differential of the dependent variable y, is defined by dy = f’(x)dx

Aim: Differentials Course: Calculus Comparing Δy and dy Let y = x 2. Find dy when x = 1 and dx = Compare this value with Δy for x = 1 and Δx = y = f(x) = x 2  f’(x) = 2x dy = f’(1)(0.01) dy = 2(0.01)= 0.02 Δy = f(c + Δx) – f(c) actual change in y dy = f’(x)dx approximate change in y dy = 2(1)(0.01)

Aim: Differentials Course: Calculus Comparing Δy and dy Let y = x 2. Find dy when x = 1 and dx = Compare this value with Δy for x = 1 and Δx = dy = f’(x)dx dy = 2(0.01)= 0.02 Δy = f(c + Δx) – f(c) actual change in yapproximate change in y Δy = f( ) – f(1) Δy = f(1.01) – f(1) Δy = – 1 2 Δy = values become closer to each other when dx or Δx approaches 0

Aim: Differentials Course: Calculus Comparing Δy and dy Let y = x 2. Find dy when x = 1 and dx = Compare this value with Δy for x = 1 and Δx = , 1 = Δx = 0.01 = 0.02

Aim: Differentials Course: Calculus Error Propagation estimations based on physical measurements A(r) = πr cm7.21 cm7.18 cm r = 7.2cm – exact measurement A(7.2) = π(7.2) 2 = A = A = A = difference is propagated error

Aim: Differentials Course: Calculus Error Propagation x + Δx Exact value f( ) Measurement error  f(x) f(x) Measurement value = Δy Propagated error Propagation error – when a measured value that has an error in measurement is used to compute another value. dy = f’(x)dx approximate change in y

Aim: Differentials Course: Calculus Model Problem The radius of a ball bearing is measured to be 0.7 inch. If the measurement is correct to within 0.01 inch, estimate the propagated error in the volume V of the ball bearing. r = 0.7measured radius < Δr < 0.01 possible error approximate ΔV by dV substitute r and dr dy = f’(x)dx approximate change in y

Aim: Differentials Course: Calculus Relative Error The radius of a ball bearing is measured to be 0.7 inch. If the measurement is correct to within 0.01 inch, estimate the propagated error in the volume V of the ball bearing. relative error  4.29%

Aim: Differentials Course: Calculus Liebniz notation Differential Formulas Let u and v be differentiable functions of x. The differential of x (denoted by dx) is any nonzero real number. The differential of y (denoted by dy) is dy = f’(x)dx. Definition  dy = y’dx

Aim: Differentials Course: Calculus Differential Formulas a.y = x 2 b.y = 2sin x c.y = xcosx d.y = 1/x FunctionDerivativeDifferential

Aim: Differentials Course: Calculus Model Problem Find the differential of composite functions y = f(x) = sin 3x y’ = f’(x) = 3cos 3x dy = f’(x)dx = 3cos 3x dx Original function Apply Chain Rule Differential Form y = f(x) = (x 2 + 1) 1/2 Original function Apply Chain Rule Differential Form

Aim: Differentials Course: Calculus Approximating Function Values Use differential to approximate x = 16 and dx = 0.5

Aim: Differentials Course: Calculus Model Problem Use differential to approximate

Aim: Differentials Course: Calculus Model Problem Find the equation of the tangent line T to the function f at the indicated point. Use this linear approximation to complete the table. x f(x)f(x) T(x)T(x)

Aim: Differentials Course: Calculus Model Problem The measurement of the side of a square is found to be 12 inches, with a possible error of 1/64 inch. Use differentials to approximate the possible propagated error in computing the area of the square.

Aim: Differentials Course: Calculus Model Problem The measurement of the radius of the end of a log is found to be 14 inches, with a possible error of ¼ inch. Use differentials to approximate the possible propagated error in computing the area of the end of the log.

Aim: Differentials Course: Calculus Model Problem The radius of a sphere is claimed to be 6 inches, with a possible error of 0.02 inch. Use differentials to approximate the maximum possible error in calculating (a) the volume of the sphere, (b) the surface area of the sphere, and (c) the relative errors in parts (a) and (b).