Suppose C(r) is the total cost of paying off a car loan borrowed at an annual interest rate of r%. What are the units of C'(r)? What is the practical meaning of C'(r)? What is its sign?

A train at a constant 60.0 km/h moves east for 40.0 min, then in a direction 50.0° east of due north for 20.0 min, and then west for 50.0 min. What are the (a) magnitude and (b) angle of its average velocity during this trip?

Prepare a poster for the CEO detailing your findings from your quadratic function investigation. Include any insights you and your teammates found. Explain your conclusions and justify your statements. Remember to include both a table and a complete graph of your parabola with all special points carefully labeled. Be thorough and complete.

Suppose the line tangent to the graph of f at x=2 is y=4x+1 and suppose the line tangent to the graph of g at x=2 has slope 3 and passes through (0, -2). Find an equation of the line tangent to the following curves at x=2. y=4f(x)

Find the general solution on the interval $(0, \infty)$ of the given DE using variation of parameters. $y^{\prime \prime}+4 y^{\prime}+4 y=\frac{e^{-2 x}}{x^{2}}$

Show that the velocity field of a line source of strength 2m can be found by integrating the (three-dimensional) velocity field of a point source of strength m dz at (0, 0, z) over the whole z-axis. Why doe s the integral correspond to a line source of strength 2m rather than strength m? Can the potential of the line source be obtained by integrating the potentials of the point sources?

Suppose f is twice differentiable on an interval I (i.e. , f " exists on I). Suppose that the points 0 and 2 belong to I and that $f(0)=f(1)=0 \text { and } f(2)=1$. Prove that

$f^{\prime}(a)=\frac{1}{2}$ for some point a in I

The quantity, Q mg, of nicotine in the body t minutes after a cigarette is smoked is given by Q=f(t). Interpret the statements f(20)=0.36 and f'(20)=-0.002 in terms of nicotine. What are the units of the numbers 20, 0.36, and -0.002?

The temperature, H, in degrees Celsius, of a cup of coffee placed on the kitchen counter is given by H = f(t), where t is in minutes since the coffee was put on the counter. (a) Is f'(t) positive or negative? Give a reason for your answer. (b) What are the units of f'(20)? What is its practical meaning in terms of the temperature of the coffee?

A point charge q1 = -4.00 nC is at the point x = 0.60 m, y = 0.80 m , and a second point charge q2 = +6.00 nC is at the point x = 0.60 m , y = 0.

a) Calculate the magnitude of the net electric field at the origin due to these two point charges. b)Calculate the direction of the net electric field at the origin due to these two point charges.

Find $c$ using the Pythagorean Theorem.

Consider an object on an incline where friction is present. The angle between the incline and the horizontal is θ and the coefficient of kinetic friction is μk. Find an expression for the acceleration of this object. Treat down the incline as the positive direction. a=?

Suppose that f(t) is a function with f(25)=3.6 and f'(25)=-0.2. Estimate f(26) and f(30).

The temperature, $H$, in degrees Celsius, of a cup of coffee placed on the kitchen counter is given by $H=f(t)$, where $t$ is in minutes since the coffee was put on the counter. (a) Is $f^{\prime}(t)$ positive or negative? Give a reason for your answer. (b) What are the units of $f^{\prime}(20)$ ? What is its practical meaning in terms of the temperature of the coffee?