Differentiation — Quiz 4
Practice Differentiation — Quiz 4 on The School of Mathematics (Differentiation) with 10 scored questions. This set covers problems such as: “Determine the value of \frac{dy}{dx} if x=1 for e^{xy}+\sin y=2 .”; “Write the equation of the tangent line to y=\frac{x^2+3}{2x+1} at x=1 .”; “At what x -values does (2x^2-x)(x+2)^{3/5} have horizontal tangent lines?”. Work carefully, check explanations for misses, then retake to track improvement before test day.
Questions in this quiz (10)
Determine the value of $$\frac{dy}{dx}$$ if $$x=1$$ for $$e^{xy}+\sin y=2$$.
- $$1$$
- $$0$$
- $$-1$$
- $$2$$
Write the equation of the tangent line to $$y=\frac{x^2+3}{2x+1}$$ at $$x=1$$.
- $$y=\frac{4}{3}-\frac{2}{9}(x-1)$$
- $$y-\frac{4}{3}=\frac{2}{9}(x-1)$$
- $$y-\frac{4}{3}=-\frac{2}{9}(x-1)$$
- $$y=-\frac{2}{9}x+\frac{4}{3}$$
At what $$x$$-values does $$(2x^2-x)(x+2)^{3/5}$$ have horizontal tangent lines?
- $$x=1$$
- $$x=-2$$
- $$x=2$$
- $$x=0,\frac12$$
If $$x=t^3+2$$ and $$y=3t^2$$, then $$\frac{dy}{dx}=$$
- $$2t$$
- $$\frac{2}{t}$$
- $$6t$$
- $$\frac{6t}{3t^2+2}$$
If $$x=t^2+2t$$ and $$y=t^3-t$$, then when $$t=1$$, $$\frac{d^2y}{dx^2}$$ is
- $$0$$
- $$1$$
- $$2$$
- $$-1$$
If $$x=e^\theta\sin\theta$$ and $$y=e^\theta\cos\theta$$, then when $$\theta=\frac{\pi}{2}$$, $$\frac{dy}{dx}$$ is
- $$0$$
- $$1$$
- $$-1$$
- $$e^{\pi/2}$$
The graph in the xy-plane represented by $$x=2+3\sin t$$ and $$y=4\cos t+1$$, for $$-\pi\le t\le\pi$$, is
- circle
- ellipse
- parabola
- half ellipse
$$\lim_{x\to0} x\sin\left(\frac{1}{x}\right)$$
- $$0$$
- $$1$$
- $$\infty$$
- oscillates
Choose the alternative that is the derivative $$\frac{dy}{dx}$$. If $$x=\frac12 t$$ and $$y=\ln(2-t)+3\ (t<2)$$.
- $$\frac12 t$$
- $$2-t$$
- $$\frac{1}{(2-t)^2}$$
- $$(2-t)^2$$
Choose the alternative that is the derivative $$\frac{dy}{dx}$$. If $$x=\frac13 t$$ and $$y=2-\ln(3-t)\ (t<3)$$.
- $$\frac13 t$$
- $$3-t$$
- $$\frac{1}{(3-t)^2}$$
- $$(3-t)^2$$