Unit 4 AP Calculus AB: Contextual Applications of Differentiation

Motion

L'Hospital

Related Rates

Local Linear Approximation

Unit 4 Quick Facts

100

A particle moves along the x-axis. The function x(t) gives the particle's position at any time t≥0

x(t)=t⁴-2t²-4

What is the particle's velocity v(t) at t=1?

A.) 3

B.) 0

C.) 1

D.) 2

B.) 0

100

Evaluate lim x->2 x^3 - 7x^2 + 10x/ x^2 + x - 6

B.) -6/5

100

A rectangle has width w inches and height h inches, where the width is twice the height. Both w and h are functions of time t, measured in seconds. If A represents the area of the rectangle, which of the following gives the rate of change of A with respect to t ? (Given: A = wh)

A.) dA/dt = 4h in/sec

B.) dA/dt = 3h in^2/sec

C.) dA/dt = 4h dh/dt in^2/sec

D.) 4h dh/dt in/sec

C.) dA/dt = 4h dh/dt in^2/sec

100

Let g be a differentiable function with g(-1) = 5 and g'(-1) = 2.

What is the value of the approximation of g(-0.9) using the function's local linear approximation at x = -1?

A.) 5.4

B.) 5.2

C.) 5

D.) 5.1

B.) 5.2

100

How do we know when to use L'hospital rule?

A.) When we need to find the derivative of a function

B.) When we need to find the local linear approximation of a function

C.) When the direct substitution of a limit gives us an indeterminate form

D.) When we need to find the slope

C.) When the direct substitution of a limit gives us an indeterminate form

200

A particle moves along the x-axis. The graph of the particle’s velocity v(t) at time t is shown above for 0<t<4.5. How many times does the particle change direction over the time interval 0<t<4.5?

A.) One time

B.) Three times

C.) Six times

D.) Two times

200

Find lim x->0 sin(x)cos(x)/x+sin(2x)

D.) 1/3

200

Charles’s law states that if the pressure of a dry gas is held constant, then the volume V of the gas and its temperature T, measured in degrees Kelvin, satisfy the relationship V=cT, where c is a constant.

Which of the following best describes the relationship between the rate of change, with respect to time t, of the volume and the rate of change, with respect to time t, of the temperature?

A.) dV/dt =T dc/dt

B.) dV/dt = c dT/dt

C.) dV/dt = c

D.) 1/t = c dT/dV

B.) dV/dt = c dT/dt

200

The local linear approximation to the function g at x = 6 is y = -3x + 4.

What is the value of g(6) + g'(6)?

A.) -17

B.) -18

C.) 17

D.) 20

A.) -17

200

What does it mean when velocity and acceleration have opposite signs?

A.) The speed is decreasing

B.) The speed is constant

C.) They cancel each other out

D.) None of the above

A.) The speed is decreasing

300

(CALCULATOR ALLOWED)

A particle moves along the y-axis so that at time t≥0 its position is given by y(t) = t^3 - 6t^2 + 9t. Over the time interval 0 < t < 4, for what values of t is the speed of the particle increasing.

A.) 2 < t < 4

B.) 3 < t < 4

C.) 0 < t < 1 and 3 < t < 4

D.) 1 < t < 2 and 3 < t < 4

300

Find lim x-> infinity 7x^2/x-x^3

B.) 0

300

A particle moves on the circle x²+y²=100 in the xy-plane for time t≥0. At the time when the particle is at the point (8,6), the value of dx/dt is 5. What is the value of dy/dt at this time?

A.) 21/3

B.) 16/6

C.) -10/3

D.) -20/3

300

No Calc

The locally linear approximation of the differentiable function f at x=3 is used to approximate the value of f(3.2). The approximation at x=3.2 is an overestimate of the corresponding function value at x=3.2. Which of the following could be the graph of f?

D.)

300

How do you know if the speed of a particle is increasing?

A.) When velocity = 0

B.) When velocity and acceleration have opposite signs

C.) When velocity and acceleration have the same signs

D.) When velocity is positive

C.) When velocity and acceleration have the same signs

400

A particle moves along the y-axis so that at time t≥0 its position is given by y(t)=(2/3t³)-5t²+8t. Over the time interval 0<t<5, for what values of t is the speed of the particle increasing?

A.) 0<t<1 and 4<t<5

B.) 0<t<1

C.) 4<t<5

D.) None of the above

A.) 0<t<1 and 4<t<5

400

Find lim x-> pi/2 -cosx /(x-pi/2)

C.) 1

400

The radius r of a sphere is increasing at a rate of 0.3 inches per second. At the instant when the surface area S becomes 100pi square inches, what is the rate of increase, in cubic inches per second in the volume V?

(Given: S = 4(pi)r^2 and V = 4/3 (pi)r^3)

A.) 10 pi

B.) 22.5 pi

C.) 50 pi

D.) 30 pi

400

(CALCULATOR ALLOWED)

The derivative of the function A is given by A'(t)=2+9e^0.4sint, and A(1.2)=7.5. If the linear approximation to A(t) at t=1.2 is used to estimate A(t), at what value of t does the linear approximation estimate that A(t)=15 ?

A.) 0.497

B.) 1.165

C.) 1.698

D.) 2.420

C.) 1.698

400

What is the formula used to find the local linear approximation of a function?

A.) y - y1 = m(x - x1)

B.) L(x) = f'(a)(x-a)+f(a)

C.) dy/dx = (x-a) + u(a)

B.) L(x) = f'(a)(x-a)+f(a)

500

Estimate Tina's acceleration at t=10. Indicate proper units in your answer.

A.) 5 meters/min

B.) 50 meters/min

C.) 5 meters/min/min

D.) 50 meters/min/min

C.) 5 meters/min/min

500

Which of the following limits does not yield an indeterminate form?

C.) lim x-> pi (x-pi/cosx)

500

(CALCULATOR ALLOWED)

The volume of a cone of radius r and height h is given by V = 1/3(pi)r^2*h. If the radius and the height both increase at a constant rate of 1/2 centimeters per second, at what rate, in cubic centimeters per second, is the volume increasing when the height is 9 centimeters and the radius is 6 centimeters.

A.) 72 pi

B.) 24 pi

C.) 20 pi

D.) 54 pi

B.) 24 pi

500

The line tangent to the graph of the twice-differentiable function f at the point x=3 is used to approximate the value of f(3.25). Which of the following statements guarantees that the tangent line approximation at x=3.25 is an underestimate of f(3.25)?

A.) The function f is decreasing on the interval 3≤x≤3.25

B.) The function f is increasing on the interval 3≤x≤3.25

C.) The graph of the function f is concave down on the interval 3≤x≤3.25

D.) The graph of the function f is concave up on the interval 3≤x≤3.25

500

When is the tangent line approximation an underestimate?

A.) When the function is decreasing

B.) When the function is concave down

C.) When the tangent line is decreasing

D.) When the function is concave up