area below x-axis is Formal definition of derivative Particle is moving to the right/up area above x-axis is

Formal definition of derivative

limit as x approaches a of [f(x)-f(a)]/(x-a) y = sin(x), y' = 6th degree Taylor Polynomial length of parametric curve Alternate definition of derivative

Alternate definition of derivative

increasing When f '(x) is positive, f(x) is given rate equation, R(t) and inital condition when t = a, R(t) = y₁ find final value when t = b Product rule Derivatives Average Rate of Change

When f '(x) is positive, f(x) is

decreasing When f '(x) is negative, f(x) is length of curve Linearization ratio test

When f '(x) is negative, f(x) is

relative minimum When f '(x) changes from negative to positive, f(x) has a y = a^x, y' = area between two curves volume of solid of revolution - no washer

When f '(x) changes from negative to positive, f(x) has a

relative maximum y = tan⁻¹(x), y' = When f '(x) changes from positive to negative, f(x) has a mean value theorem When f '(x) is increasing, f(x) is

When f '(x) changes from positive to negative, f(x) has a

concave up dP/dt = kP(M - P) methods of integration When f '(x) changes from increasing to decreasing or decreasing to increasing, f(x) has a When f '(x) is increasing, f(x) is

When f '(x) is increasing, f(x) is

infinite jump removable d/du e^u Circle Circumference Product Rule situations where limit fails to exist

situations where limit fails to exist

lim h->0 f(x+h)-f(x)/h situations where limit fails to exist chain rule formal definition of a derivative d/du arccosu

formal definition of a derivative

lim x->a f(x)-f(a)/x-a d/du arcsinu d/du log base a of u Circle Circumference Alternate form of definition of a derivative

Alternate form of definition of a derivative

1) discontinuity 2) cusp 3) vertical tangent line Situations Derivatives Fail to Exist d/du log base a of u chain rule formal definition of a derivative

Situations Derivatives Fail to Exist

Let f be continuous on [a,b] and differentiable on (a,b) and if f(a)=f(b) then there is at least one number c on (a,b) such that f'(c)=0 (If the slope of the secant is 0, the derivative must = 0 somewhere in the interval). integral scaling rule Fundamental Theorem of Calculus FTC Rolle's Theorem, Graf funkcije gre od točke (a,f(a))(a, f(a))(a,f(a)) do točke (b,f(b))(b, f(b))(b,f(b)), ki sta na isti višini.Če je funkcija gladka (zvezna in odvedljiva), potem mora nekje vmes "obrniti smer" — točka, kjer je odvod enak 0 (vodoravna tangenta).

Rolle's Theorem, Graf funkcije gre od točke (a,f(a))(a, f(a))(a,f(a)) do točke (b,f(b))(b, f(b))(b,f(b)), ki sta na isti višini.Če je funkcija gladka (zvezna in odvedljiva), potem mora nekje vmes "obrniti smer" — točka, kjer je odvod enak 0 (vodoravna tangenta).

E. x=0, or y=0, or y=-x If f(x)= x^3-x-6 for all real numbers x, and if g is the inverse function of f, then f'(g(0))g'(0) = What are all values of x for which the graph of y=6x^2+x/2+3+6/x is concave downward At what point on the curve x^3+3x^2+y^2=4 is the tangent line vertical? The slope field for the differential equation dy/dx=(x^2y+y^2x)/(3x+y) will have horizontal segments when

The slope field for the differential equation dy/dx=(x^2y+y^2x)/(3x+y) will have horizontal segments when

A. -2 If f'(x)=3x^2 and f(-1)=2 then the integral from 0 to 2 of f(x) dx If C(x) gives the cost in dollars of producing x items of a certain product, which of the following statements are true about dC/dx, the derivative of C(x) the integral of csc^2xdx If f(x) = (x-1)^2cosx, then f'(0)=

If f(x) = (x-1)^2cosx, then f'(0)=

A. y+25x=32 If f(x)=2x^3 then the average rate of change of f on the interval [0,2] is On the interval a<x<b, the function f is positive, decreasing, and concave upwards. Which of the following has the greatest value? If f(x)=e^xlnx, then f'(e)= An equation of the line tangent to the curve y= (3x+4)/4x-3) at the point (1,7) is

An equation of the line tangent to the curve y= (3x+4)/4x-3) at the point (1,7) is

E. 7/2 A particle moves along the x-axis so that at any time t its position is given by x(t)=(1/2)sin t + cos(2t). What is the acceleration of the particle at t= pi/2 What is the area of the largest rectangle with lower base on the x-axis and upper vertices on the curve y=12-x^2 If dy/dx= ycosx and y=3 when x=0, then y = A missile rises vertically from a point on the ground 75000 feet from a radar station. If the missile is rising at the rate of 16,500 feet per minute at the instant when it is 38,000 feet high, what is the rat e of change, in radians per minute, of the missile's angle of elevation from the radar station at this instant?

A particle moves along the x-axis so that at any time t its position is given by x(t)=(1/2)sin t + cos(2t). What is the acceleration of the particle at t= pi/2

D. 1/2 the integral from 0 to 3 of x/sqrt(x^2+16)= lim as x approaches positive infinity of (x-1/2x)/(2x+1/6x) is On the interval a<x<b, the function f is positive, decreasing, and concave upwards. Which of the following has the greatest value? Let f be, the function defined by f(x)=ln(3x+2)^k for some positive constant k. If f'(2)=3, what is the value of k?

lim as x approaches positive infinity of (x-1/2x)/(2x+1/6x) is

C. 7 If the line y=4x+3 is tangent to the curve y=x^2+c, then c is Let f be, the function defined by f(x)=ln(3x+2)^k for some positive constant k. If f'(2)=3, what is the value of k? At what point on the curve x^3+3x^2+y^2=4 is the tangent line vertical? An equation of the line tangent to the curve y= (3x+4)/4x-3) at the point (1,7) is

If the line y=4x+3 is tangent to the curve y=x^2+c, then c is

A. 1 If the integral from 0 to 1000 8^xdx - the integral from a to 1000 8^xdx= 10.40, the a= If dy/dx= ycosx and y=3 when x=0, then y = the integral from 0 to 3 of x/sqrt(x^2+16)= If f(t)>0 for all t, and the integral from a to b of f'(t)dt/f(t)=ln[f(b)], then f(a) is

the integral from 0 to 3 of x/sqrt(x^2+16)=

B. 1/2 If f(x)=e^xlnx, then f'(e)= lim as x approaches positive infinity of (x-1/2x)/(2x+1/6x) is What are all values of x for which the graph of y=6x^2+x/2+3+6/x is concave downward The total area enclosed by the curves y=x^3 and y=x is

The total area enclosed by the curves y=x^3 and y=x is

Y-Intercept = (0,-467) Problem Statement: It is exactly 213 miles from Harrisburg to Aberdeen. It takes you 3 hours to get there traveling by car. Describe the y-intercept of this function in context of the situation it represents. Problem Statement: It is exactly 213 miles from Harrisburg to Aberdeen. It takes you 3 hours to get there traveling by car. Determine how far away from Aberdeen you are if you have been traveling for 2.5 hours. Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. Determine the x-intercept as an ordered pair (x,y). Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. Determine the y-intercept as an ordered pair (x,y).

Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. Determine the y-intercept as an ordered pair (x,y).

X-Intercept = (6.2,0) Problem Statement: It is exactly 213 miles from Harrisburg to Aberdeen. It takes you 3 hours to get there traveling by car. Determine an appropriate domain and range for this function. Write your answer as an inequality. Problem Statement: It is exactly 213 miles from Harrisburg to Aberdeen. It takes you 3 hours to get there traveling by car. Describe the rate of change in context of the situation it represents. Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. Write a sentence explaining what the x-intercept means in context. Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. Determine the x-intercept as an ordered pair (x,y).

Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. Determine the x-intercept as an ordered pair (x,y).

Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. Determine the Window Values (x-min, x-max, x-scl, y-min, y-max, y-scl) for this graph. Problem Statement: It is exactly 213 miles from Harrisburg to Aberdeen. It takes you 3 hours to get there traveling by car. Write a linear function that describes the distance in miles D(t) that you are away from Aberdeen as a function of time in hours, t, which you have been traveling. Problem Statement: It is exactly 213 miles from Harrisburg to Aberdeen. It takes you 3 hours to get there traveling by car. Graph your linear function with an appropriate label and scale. Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. Looking at the graph, state the appropriate domain and range for this function.

Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. Determine the Window Values (x-min, x-max, x-scl, y-min, y-max, y-scl) for this graph.

Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. Determine the Window Values (x-min, x-max, x-scl, y-min, y-max, y-scl) for this graph. Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. Use your graphing calculator to graph this function. Sketch the graph using an appropriate label and scale. Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. After 4 minutes, how far below the surface of the water will the submarine be? Problem Statement: It is exactly 213 miles from Harrisburg to Aberdeen. It takes you 3 hours to get there traveling by car. Describe the rate of change in context of the situation it represents.

Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. Use your graphing calculator to graph this function. Sketch the graph using an appropriate label and scale.

After 6.2 minutes, the submarine will be at the surface of the water (0 minutes). Problem Statement: It is exactly 213 miles from Harrisburg to Aberdeen. It takes you 3 hours to get there traveling by car. Determine how long you have been traveling (in hours) if you are 100 miles away from Aberdeen. Problem Statement: It is exactly 213 miles from Harrisburg to Aberdeen. It takes you 3 hours to get there traveling by car. Describe the x-intercept of this function in context of the situation it represents. Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. Determine the Window Values (x-min, x-max, x-scl, y-min, y-max, y-scl) for this graph. Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. Write a sentence explaining what the x-intercept means in context.

Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. Write a sentence explaining what the x-intercept means in context.

The submarine started 467 meters below the surface of the water at 0 minutes. Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. Write a sentence explaining what the y-intercept means in context. Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. Write a sentence explaining what the x-intercept means in context. Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. Looking at the graph, state the appropriate domain and range for this function. Problem Statement: It is exactly 213 miles from Harrisburg to Aberdeen. It takes you 3 hours to get there traveling by car. Describe the x-intercept of this function in context of the situation it represents.

Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. Write a sentence explaining what the y-intercept means in context.

167 meters Problem Statement: It is exactly 213 miles from Harrisburg to Aberdeen. It takes you 3 hours to get there traveling by car. Describe the x-intercept of this function in context of the situation it represents. Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. Determine the Window Values (x-min, x-max, x-scl, y-min, y-max, y-scl) for this graph. Problem Statement: It is exactly 213 miles from Harrisburg to Aberdeen. It takes you 3 hours to get there traveling by car. Write a linear function that describes the distance in miles D(t) that you are away from Aberdeen as a function of time in hours, t, which you have been traveling. Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. After 4 minutes, how far below the surface of the water will the submarine be?

Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. After 4 minutes, how far below the surface of the water will the submarine be?

4.2 minutes Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. Determine the Window Values (x-min, x-max, x-scl, y-min, y-max, y-scl) for this graph. Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. Write a sentence explaining what the x-intercept means in context. Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. After how many minutes will the submarine have a position of -150 meters? Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. After 4 minutes, how far below the surface of the water will the submarine be?

Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. After how many minutes will the submarine have a position of -150 meters?

Domain: 0 <= x <= 6.2 Range: -467 <= y <= 0 Problem Statement: It is exactly 213 miles from Harrisburg to Aberdeen. It takes you 3 hours to get there traveling by car. Determine how far away from Aberdeen you are if you have been traveling for 2.5 hours. Problem Statement: It is exactly 213 miles from Harrisburg to Aberdeen. It takes you 3 hours to get there traveling by car. Describe the rate of change in context of the situation it represents. Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. Write a sentence explaining what the y-intercept means in context. Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. Looking at the graph, state the appropriate domain and range for this function.

Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. Looking at the graph, state the appropriate domain and range for this function.

D(t) = -71t + 213 Problem Statement: It is exactly 213 miles from Harrisburg to Aberdeen. It takes you 3 hours to get there traveling by car. Write a linear function that describes the distance in miles D(t) that you are away from Aberdeen as a function of time in hours, t, which you have been traveling. Problem Statement: It is exactly 213 miles from Harrisburg to Aberdeen. It takes you 3 hours to get there traveling by car. Describe the x-intercept of this function in context of the situation it represents. Problem Statement: It is exactly 213 miles from Harrisburg to Aberdeen. It takes you 3 hours to get there traveling by car. Determine how far away from Aberdeen you are if you have been traveling for 2.5 hours. Problem Statement: A submarine starts 467 meters below the surface of the ocean. The submarine ascends at 75 meters per minute. This relationship can be modeled by the linear function D(t) = 75t - 467 where D(t) represents the position of the submarine and t is time. Determine the y-intercept as an ordered pair (x,y).

Problem Statement: It is exactly 213 miles from Harrisburg to Aberdeen. It takes you 3 hours to get there traveling by car. Write a linear function that describes the distance in miles D(t) that you are away from Aberdeen as a function of time in hours, t, which you have been traveling.