Review Derivatives When you see the words… This is what you know… f has a local (relative) minimum at x = a f(a) is less than or equal to every other y-value in some interval containing x = a 1
Review Derivatives When you see the words… This is what you know… The difference quotient for f at x = a .. 2
Review Derivatives When you see the words… This is what you know… Definition of derivative f’(x) = .. 3
Review Derivatives When you see the words… This is what you know… If f is differentiable at x = a Then f is continuous at x = a 4
Review Derivatives When you see the words… This is what you know… Rolle’s Theorem If f is continuous on [a,b], differentiable on (a,b) and f(a) = f(b), then for some c on (a,b), f’( c) = 0 5
Review Derivatives When you see the words… This is what you know… Mean Value Theorem If f is continuous on [a,b], differentiable on (a,b) then for some c on (a,b) 6
Review Derivatives When you see the words… This is what you know… Mean Value Theorem (as rates of change) If f is continuous on [a,b], differentiable on (a,b) then for some point in the interval the instantaneous rate of change = average rate of change 7
Review Derivatives When you see the words… This is what you know… If f is differentiable at x = a, then …… 8
Review Derivatives When you see the words… This is what you know… .. k * f’ 9
Review Derivatives When you see the words… This is what you know… .. f’ g’ 10
Review Derivatives When you see the words… This is what you know… .. f * g’ + g * f’ 11
Review Derivatives When you see the words… This is what you know… .. 12
Review Derivatives When you see the words… This is what you know… .. 13
Review Derivatives When you see the words… This is what you know… .. 14
Review Derivatives When you see the words… This is what you know… .. 15
Review Derivatives When you see the words… This is what you know… .. 16
Review Derivatives When you see the words… This is what you know… .. 17
Review Derivatives When you see the words… This is what you know… .. 18
Review Derivatives When you see the words… This is what you know… .. 19
Review Derivatives When you see the words… This is what you know… Average rate of change of f on [a,b] .. 20
Review Derivatives When you see the words… This is what you know… f’(a) = 0 The graph of f has a horizontal tangent at x=a 21
Review Derivatives When you see the words… This is what you know… Instantaneous rate of change of f at x = a f’(a) 22
Review Derivatives When you see the words… This is what you know… Critical points of f At endpoints of the domain Where f’(x) does not exist Where f’(x) = 0 23
Review Derivatives When you see the words… This is what you know… f’(x) > 0 for a < x < b f is increasing on the interval a < x < b 24
Review Derivatives When you see the words… This is what you know… f’(x) < 0 for a < x < b f is decreasing on the interval a < x < b 25
Review Derivatives When you see the words… This is what you know… f” (x) > 0 for a < x < b The graph of f is concave upward on the interval a < x < b 26
Review Derivatives When you see the words… This is what you know… f” (x) < 0 for a < x < b The graph of f is concave downward on the interval a < x < b 27
Review Derivatives When you see the words… This is what you know… f’ (x) is increasing for a < x < b The graph of f is concave upward on the interval a < x < b 28
Review Derivatives When you see the words… This is what you know… f’ (x) is decreasing for a < x < b The graph of f is concave downward on the interval a < x < b 29
Review Derivatives When you see the words… This is what you know… The second derivative test f’(a) = 0 and f”(a) < 0 f has a maximum at x = a 30
Review Derivatives When you see the words… This is what you know… The second derivative test f’(a) = 0 and f”(a) > 0 f has a minimum at x = a 31
Review Derivatives When you see the words… This is what you know… The graph of f changes concavity The graph of f has a point of inflection 32
Review Derivatives When you see the words… This is what you know… At x = a, f is continuous and changes from increasing to decreasing f has a maximum at x = a 33
Review Derivatives When you see the words… This is what you know… At x = a, f is continuous and changes from decreasing to increasing f has a minimum at x = a 34
Review Derivatives When you see the words… This is what you know… Velocity, speed, and acceleration Velocity: Speed: | v(t) | Accleration: 35
Review Derivatives When you see the words… This is what you know… Increasing speed Velocity and acceleration have the same sign 36
Review Derivatives When you see the words… This is what you know… Decreasing speed Velocity and acceleration have the opposite signs 37
Review Derivatives When you see the words… This is what you know… The normal line to a curve at x = a is Perpendicular to the line tangent to the curve at x = a 38
Review Derivatives When you see the words… This is what you know… An object in motion along a line reverses direction when… The sign of the object’s velocity changes 39
Review Derivatives When you see the words… This is what you know… An object is at rest when… v(t) = 0 40
Review Derivatives When you see the words… This is what you know… f has a global (absolute) maximum at x = a f(a) is greater than or equal to every other y-value of f 41
Review Derivatives When you see the words… This is what you know… f has a global (absolute) minimum at x = a f(a) is less than or equal to every other y-value of f 42
Review Derivatives When you see the words… This is what you know… f has a local (relative) maximum at x = a f(a) is greater than or equal to every other y-value in some interval containing x = a 43
Review Derivatives When you see the words… This is what you know… .. Cos x 44
Review Derivatives When you see the words… This is what you know… .. -sin x 45