If \(f(2) = 7\) and \(f'(2) = -3\), then the equation of the tangent to the curve \(y = f(x)\) at \(x = 2\) is

On the interval \(1 < x < 2\), the curve \(y = x^3 - 6 x^2 + 9 x + 1\) is

The minimum value of the function \(f(x) = \sqrt[3]{x^2 + 4 a x + 12 a^2}\), \(a > 0\), is

\(\operatorname*{lim}\limits_{x \rightarrow -\infty} \dfrac{10 - 2^x}{10 + 2^{-x}} =\)

The x-coordinate of the point where the tangent to the parabola \(y = a x^2\) at \(x = p\) (not a vertex) intersects the x-axis is

The table below shows some of the values of two differentiable functions \(f\) and \(g\) and their derivatives. If \(h(x) = f(x) g(x)\), then \(h'(5) = \)

\(x\)	\(f(x)\)	\(g(x)\)	\(f'(x)\)	\(g'(x)\)
3	-3	6	-5	1
4	0	3	-3	9
5	3	-2	4	5

Using the values in the table from the previous problem (\(f(3)=-3\), \(f(4)=0\), \(f(5)=3\), \(g(3)=6\), \(g(4)=3\), \(g(5)=-2\), \(f'(3)=-5\), \(f'(4)=-3\), \(f'(5)=4\), \(g'(3)=1\), \(g'(4)=9\), \(g'(5)=5\)), if \(h(x) = f(g(x))\), then \(h'(4) =\)

If \(f(x)\) is a continuous function and \(f(2) = 7\) and \(f'(2) = -3\), then \(f(2.01)\) is approximately

Consider the curve \(y = 2 x^3 - 3(k + 1) x^2 + 6 k x\), \(k > 1\). On the interval \(1 < x < k\),

If \(f(x) = 2^x\) and \(2^{3.03} \approx 8.168\), which of the following is closest to \(f'(3)\)?

Pictured above (in source) is the graph of \(f'(x)\), which is positive on \((-4, 4)\) and reaches its maximum at \(x = 0\). For what values of \(x\) is the graph of \(f(x)\) concave down?

If \(g(1) = 3\), \(g'(1) = 4\), \(g(2) = 8\), and \(g'(2) = 3\), and \(f(x) = g^2(x)\), then \(f'(2) =\)

If \(\displaystyle\int_{0}^{4} f(x) d x = 10\), \(\displaystyle\int_{0}^{5} f(x) d x = 9\), and \(\displaystyle\int_{4}^{7} f(x) d x = 1\), then \(\displaystyle\int_{5}^{7} f(x) d x =\)

If \(u = x^2 + 1\), then \(\displaystyle\int_{1}^{2} \dfrac{x^2}{x^2 + 1} d x =\)

The average area of all circles with radii between 3 and 6 is

A rumor spreads continuously at the rate of \(3 t^2 + 6 t\) (where \(t\) is measured in days). How many people hear the rumor on the third day?

If \(\operatorname*{lim}\limits_{x \rightarrow 2}[\ln f(x)] = 1\), then \(\operatorname*{lim}\limits_{x \rightarrow 2} f(x) =\)

The graph of \(y = f''(x)\) consists of two straight line segments. It passes through \((0, 3)\), \((3, 0)\), \((6, -3)\), \((9, 0)\). If \(f'(0) = 0\), then in the vicinity of which of the following values of \(x\) is the curve \(y = f(x)\) falling and concave down?

If \(f(x) = \sin 2 x \cos 3 x\) and \(k\) is an odd integer, then \(f'(k \pi) =\)

If \(F(x) = \displaystyle\int_{1}^{x} \dfrac{4}{1 + \ln t} d t\), then \(F'(e) =\)

If the slope of the tangent to the curve at any point \((x, y)\) on the curve equals \(\dfrac{x}{y}\), what kind of curve can it be?