Calculus Studio

Final round

Ten problems from the whole studio. Nothing tells you which idea to use. 0 of 10 solved

school

A cyclist rides 30 km at 20 km/h, then another 30 km at 30 km/h. What is her average speed for the whole ride, in km/h?

school

\lim_{h\to 0} \dfrac{\frac{1}{2+h} - \frac{1}{2}}{h}

school

The tangent to $y = x^3$ at x = 1 is extended until it meets the x-axis. At what x does it meet it?

school

What is the smallest value of $f(x) = 2x^3 - 3x^2 - 12x$ for x between 0 and 3?

high school

Two positive numbers x and y add up to 12. What is the largest possible value of $x\,y^2$ ?

high school

A circle's area grows at 10π cm² per second. How fast is its radius growing when the radius is 5 cm, in cm per second?

high school

Use the tangent line to y = ∛x at x = 8 to estimate ∛8.12. Give two decimal places.

high school

\int_0^2 \left(3x^2 + 1\right)\,dx

high school

What is the average value of $x^2$ for x from 0 to 3?

high school

A culture grows at a rate proportional to its size. It goes from 100 cells to 300 cells in 2 hours. How many cells are there after 6 hours?