Review Midterm Calculus

Front

When is the function f(x)=x^2+6x+9 increasing?

Back

The function f(x)=x^2+6x+9 is increasing on the interval (-3, ∞). This is determined by finding the derivative and setting it to zero to find critical points.

Front

What is the formula for the area A of a circle in terms of its radius r?

Back

The area A of a circle is given by the formula A = πr^2.

Front

If the radius r of a circle is increasing at a rate of 5 ft per hour, what is the rate of change of the area when r = 10 ft?

Back

The rate of change of the area is 100π square feet per hour when the radius is 10 ft.

Front

What is the average rate of change of the function f(x)=x^2-6x+6 from x=2 to x=6?

Back

The average rate of change from x=2 to x=6 is 2.

Front

What is the derivative of the function f(x) = xe^x?

Back

The derivative f'(x) = e^x + xe^x.

Front

What is the acceleration of an object if its position is given by x = 3t^2 + 5t^3 - 2t?

Back

The acceleration of the object at time t = 2 s is 66 m/s².

Front

What is the first derivative of a function used to determine?

Back

The first derivative of a function gives the rate of change of the function, indicating whether it is increasing or decreasing.

Front

How do you find the critical points of a function?

Back

To find the critical points, take the derivative of the function, set it equal to zero, and solve for x.

Front

What does the second derivative of a function indicate?

Back

The second derivative indicates the concavity of the function and can be used to determine points of inflection.

Front

What is the relationship between the radius and the area of a circle?

Back

The area of a circle increases with the square of the radius, meaning if the radius doubles, the area increases by a factor of four.