Pt 2: AB Practice Test

what is the limit of -∞/a?

-∞

what is the limit of a/-∞?

0

what is the limit of a/∞?

0

formula for (f⁻¹)'(x)

1 / f'(f⁻¹(x))

y³-xy=2 and dy/dx= y / 3y²-x 1) write an equation for the line tangent to curve at (-1,1) 2) find coordinates of point where line tangent to curve is vertical

1) a)find m dy/dx = slope of curve = m m = 1 / 3(1)²-1 m = 1/4 b) y-y₁=m(x-x₁) y = 1/4(x+1) + 1 2) vertical when denom of dy/dx is 0 a) find x in terms of y 3y²-x = 0 x=3y² b) find y plug in x in terms of y (*into OG equation*) y³-xy = 2 y³ - (3y²)(y) = 2 y=-1 c) find x plug y into OG equation x=3 ANSWER: (3,-1)

does trapezoidal sum over or underestimate?: 1) concave up graph 2) concave down graph

1) concave up: overestimate 2) concave down: underestimate

when can you use Extreme Value Theorem?

1) continuous 2) closed interval

if f'(x)>0 for all real numbers x and ∫⁷₄ f(t) dt = 0, how could you find a table of data for x and f(x)?

1) f'(x) is positive = f(x) is increasing 2) if ∫=0 there must be both + and - values of f(x)

The maximum acceleration attained on the interval 0≤ t ≤3 by the particle whose velocity is given by v(t) = t³-3t²+12t+4 is

1) find a'(t) 2) set = 0 (CV) 3) plug CVs and endpoints into a(t) and find which is biggest ANSWER: 21

how to find rate that person's shadow is lengthening (streetlight problem)

1) label streetlight and person height 2) x=distance between light and person; l= distance between person and edge of triangle 3) RATIO: visualize smaller triangle made by person inside whole triangle (big triangle height is light and base is x+l; small triangle height is person and base is l) set ratio: (x+l)/l = lamp height/person height 4) get l by itself 5) take IMPLICIT differentiation (plug in dx/dt given in problem) dl/dt = ANSWER

steps for optimization problem (min or max value of something)

1) make equation (using x and relating all unknowns to it) 2) define interval 3) take d/dx 4) find CVs 5) plug in CVs and endpoints (interval) into OG equation MAKE SURE CV within interval

f is continuous on [1,3] with f(1)=10 and f(3)=18. Which of the following must be true? A) 10≤f(2)≤18 B) f(x)=17 has at least one solution in interval [1,3] C) all of the above

B) (must reach 17 to be continuous; A is wrong bc continuous function could be parabola and go - before coming back up to 18; choose LESS SPECIFIC answer)

you are given a table with the points of x and f'(x): (-2,3) ; (0,1) ; (3,4) ; (5,7) ; (6,5) which of the following must be true: A) graph of f is concave up B) graph of f has at least two POI C) f is increasing D) f has no critical points E) f has at least two relative extrema

B) f'(x) goes from decrease, increase, decrease (NOT C bc you don't know if negative #s between)

when there is a vertical tangent what does that mean about f'(x) at that spot?

DNE slope is 0

the line y=5 is a horizontal asymptote to the graph of which of the following? A) y=sin(5x) / x B) y=5x C) y=1 / x-5 D) y= 5x / 1-x E) y = 20x²-x / 1+4x²

E) plug in x as ∞ A) sin only goes from 1 to -1 (no) B) (no) C) (no) D) 1 is negligible; (no) E) x and 1 are negligible; 20x²/4x² = 5

lim h→0 : [ln(4+h) - ln(4)] / h

L'Hospital: [(1 / 4+h) - 0(constant)] / 1 1/4

f(x) is concave up on (0,2) and is tangent to y=3x-2 at x=1. a) f'(x) ≤ 3 on (0.9,1) b) f'(x) ≥ 3 on (0.9,1)

a) according to graph it would be flatter just before 1

lim (as n→0): (e⁻¹⁻ⁿ - e⁻¹) / n

d/dx of num: e⁻¹⁻ⁿ( -1) / 1 (e⁻¹ is a constant so disappears) ANSWER: -1/e

If dy/dt = ky and k is a nonzero constant, then y could be

dy = ky dt ∫1/y dy = ∫k dt lny = kt + C y = e^kt + C y = e^kt (e^c) y = A (e^kt)

the mean value theorem only applies under what 3 conditions?

f(x) is 1) continuous 2) differentiable 3) closed interval

find dy/dx if y= (x / x+1)⁵

f= ( )⁵ f' = 5( )⁴ g = x / x+1 g' = 1 / (x+1) ² ANSWER: 5x⁴ / (x+1)⁶

T or F? relative extremes can only occur on closed interval

false

how would you find the max value of f(x) given the graph of f'(x)

find max value of x (change from + to - on f') f(0)+area under curve up to max x

g(x)>0 and f(0)=1 h(x) = f(x)g(x) and h'(x)= f(x)g'(x) what does f(x) =?

h'(x) = f'(x)(g(x)) + g'(x)(f(x)) f'(x)(g(x))=0 f'(x)=0 f(x) = a constant answer choice

how would you find the area under a curve if it is half of an ellipse

half of ellipse formula: πab

rewrite ln(1) - ln(c)

ln (1/c)

whenever you are taking a d/dx and you see a y

multiply d/dx by *(dy/dx)*

when asked which of the following could be false about continuous, differentiable function on interval look for...

one that doesn't follow a theorem (esp Rolle's theorem that f'(x) = 0)

how to determine the limit of a piecewise function with 2 equations on different but overlapping intervals

plug in # from limit→# see if limit can exist based on equations (ie if one is ln2 and one is 4ln2, cannot exist)

given the coordinates of the corners of a rectangle (with x's in coordinates), how would you determine the value of x that makes rectangle have max area?

plug in values given for x determine what length and width are for each x multiply to find biggest area

how could you determine which equations of f(x) and g(x) make the limit as x→∞ of f(x)/g(x) infinite?

plug in ∞ and use rules

what does the mean value theorem equation mean?

slope of tangent/derivative (instantaneous rate of change) = slope of secant (average rate of change)

how would you solve for y if -1/y = x²/2 - 3

take negative reciprocal ALL AT ONCE y = -1 / (x²/2 - 3)

what is the limit of ∞/a?

∞

f is a linear function. ∫f"(x) =

∫f"(x) = f'(x) linear slope = 0

f(x) = {2 for x<3 {x-1 for x≥3 What is ∫₁⁵ f(x) dx?

∫₁³ 2dx + ∫₃⁵ x-1 *include 3 in both limits* ANSWER: 10

if P(t) is size of population at time t, what equation could describe linear growth in size of population? dP/dt = ?

any constant number (if linear, slope is constant)

if f(2)=6 a) lim as x→2: f(2x)=12 b) lim as x→2: (f(x))² = 36 c) all of the above

b) (NOT a bc you do not know that f(4)=12; slope may not be linear)

intermediate value theorem only applies to

continuous functions

what is the limit of -∞/-∞?

use L'Hospitals rule

what is the limit of 0/0?

use L'Hospitals rule

what is the limit of ∞/∞?

use L'Hospitals rule