Consider the two functions shown below. Assume the blue line represents a(x) and the pink line represents b(x). Which statement is true about these functions at x=3?

Find the derivative of U(x) with respect to x for U(x)=4x²-3x+2.

dU/dx=8x-3

On the plot shown, assume the horizontal axis represents x and the vertical axis represents y. At what value of x does the function y(x) have a value of zero and a positive derivative?

The function y(x) is zero at x = 2 and at x = 6

But

at x = 2, the function is decresing with negative slope===> negative derivative

while

at x = 6, the function is incresing with positive slope===> positive derivative

We know that the velocity (v(t)) is the derivative of position (x(t)) with respect to time, meaning v(t) = dx/dt. Given that, what do we get if we integrate the velocity of an object from t=1 to t=4, meaning integral t=4 t=1 v(t)dt?

The change in the object’s position between t=1 and t=4

What is the derivative of B(t) with respect to t for B(t) = Asin(kx-omegat)?

dB/dt=-omegaAcos(kx-omegat)

Find the derivative of y(x) with respect to x for y(x)=4+3x.

dy/dx=3

The potential energy of a particle experiencing a certain kind of force is given by U(x)=2x+(8/x) measured in joules (J), for positive values of x. What is the minimum potential energy this particle can have?

8 J

What is the derivative of U(x) with respect to x for U(x)=10e^5x?

dU/dx=50e^5x

What is the correct mathematical translation of the following sentence: “P is the derivative of W with respect to time (t).”?

P=dW/dt

What is the rate of change of cos(x) with respect to x?

−

On the plot shown, assume the horizontal axis represents x and the vertical axis represents y. What is true about the derivative of the function y(x) at x = 5?

The derivative of y(x) at x=5 is positive.

The rate of change of quantity Z is also called its ______.

derivative

Explanation:

The rate of change of quantity Z with respect to time is also called its derivative.

The integral of a quantity represents summing up the area under the function, not its rate of change.

A cross product is a method of multiplying two vectors and is not a rate of change.

A dot product is a method of multiplying two vectors and is not a rate of change.

Which is the correct mathematical representation of the area indicated in the figure?

The area under a curve is the integral of the function over the indicated range.

Find the derivative of f(t) with respect to t for f(t)=10t¹⁰.

Differentiate using the power rule, d/dx[xⁿ]=nx^n-1.

df/dt=10 x 10 t ^(10-1)

df/dt=100t⁹

What is d/dx sin(x)?

cos(x)

The force exerted on a particle experiencing a certain kind of force is given by F(t)=-3t⁴+8t³+5, with F measured in newtons (N) and t measured in s. At what time is the most force exerted on the particle?

At about t=2 seconds

Consider the two functions shown below. Assume the blue line represents a(x) and the pink line represents b(x). Which statement is true about these functions at x=4.5?

What is the derivative of E(x) with respect to x for E(x)=Acos(kx)?

dE/dx=-kAsin(kx)

On the plot shown, assume the horizontal axis represents x and the vertical axis represents y. At what value of x does the function y(x) have a non-zero value and a zero derivative?

At x=4

Find the derivative of a(y) with respect to y for a(y)=2/y² (1+(2y³/5)-(5/y⁴)).

da/dy=(-4/y³)+(4/5)+(60/y⁷)

What is the correct mathematical translation of the following sentence: “W is the integral of F with respect to distance (x).”?

Consider the two functions shown below. Assume the blue line represents a(x) and the pink line represents b(x). How do the second derivatives of the two functions with respect to x compare?

At what value of t will the function v(t)=-3/t² be parallel to the function omega(t)=(3t/4)-5?

t=2

Find the derivative of y(x) with respect to x for y(x)=4+3x.

dy/dx=3

What is the derivative of v(t) with respect to t for v(t)=3e^t.

dv/dt=v(t)=3e^t

Find the derivative of B(t) with respect to t for

B(t) = 1/t³ - 1/t⁵ + 1/t
B'(t) = d (1/t³ - 1/t⁵ + 1/t) / dt

= -3/t⁴ + 5/t⁶ -1/t² (d(xⁿ)/dx = n x^(n-1) ) n is constant

dB/dt = = (-3/t⁴)+ (5/t⁶)-(1/t²)