Unit 3 AP Calculus AB review

Let f be the function $$f\left(x\right)=x^2\ln x$$ . For what value of x is the slope of the tangent line to the graph of f at $$\left(x,f\left(x\right)\right)$$ equal to 3?

Let f be the function given by $$f\left(x\right)=4e^{2x}$$ and let g be the function given by $$g\left(x\right)=3x^3$$ . At what value of x do the graphs of f and g have parallel tangent lines?

A particle moves along a line so that at time t, where $$0\le t\le\pi$$ , its position is given by $$s\left(t\right)=-4\sin t-\frac{t^2}{2}+15$$ . What is the speed of the particle when its acceleration is zero?

Consider the curve $$x^2y+y^2x=6$$

a. Write the equation for the tangent line where 𝑥 = 2 and 𝑦 = 1.

b. Find the coordinates of the point (𝑥, 𝑦) where the curve has a horizontal tangent line.