Differentiation — Quiz 3

Questions in this quiz (10)

1. Evaluate $$h'(2)$$ if $$h(x)=3x\log_3 x$$.

   • $$3\log_3 2$$
   • $$\frac{3}{\ln 3}$$
   • $$3\log_3 2+\frac{3}{\ln 3}$$
   • $$6\log_3 2$$

2. Given functions $$f$$ and $$g$$ such that $$f(2)=4,\ f'(2)=1,\ g(2)=3,\ g'(2)=5$$. If $$k(x)=g(x)\sqrt{f(x)}$$, evaluate $$k'(2)$$.

   • $$11$$
   • $$\frac{43}{4}$$
   • $$9$$
   • $$13$$

3. Find $$p'(t)$$ if $$p(t)=\frac{4t^3}{\cot 3t}$$.

   • $$\frac{12t^2\cot 3t+12t^3\csc^2 3t}{\cot^2 3t}$$
   • $$\frac{12t^2}{\cot 3t}+4t^3\csc^2 3t$$
   • $$\frac{12t^2\cot 3t-12t^3\csc^2 3t}{\cot^2 3t}$$
   • $$\frac{12t^2}{\cot 3t}-4t^3\csc^2 3t$$

4. Find $$f'(1)$$ if $$f(x)=\frac{x^3\ln x}{2x^2+1}$$.

   • $$0$$
   • $$\frac{1}{3}$$
   • $$1$$
   • $$2$$

5. If $$f(x)$$ and $$g(x)$$ are differentiable, find $$h'(1)$$ if $$h(x)=\frac{2f(4x)}{g(x)}$$. Use the table values.

   • $$10.56$$
   • $$8.21$$
   • $$6.75$$
   • $$12.11$$

6. Find the equation of the tangent line to $$x^2+y^2=25$$ when $$x=3$$.

   • $$y-4=-\frac{3}{4}(x-3)$$
   • $$y-4=\frac{4}{3}(x-3)$$
   • $$y+4=-\frac{3}{4}(x-3)$$
   • $$y=-\frac{3}{4}x+4$$

7. Find the equation of the normal line to $$\cos(x)+e^{xy}=2$$ when $$x=\pi$$.

   • Normal line expression
   • Chain rule only
   • 0
   • 1

8. Find $$\frac{dy}{dx}$$ if $$e^{2y}+4y=\tan x$$.

   • $$\frac{\sec^2 x}{2e^{2y}+4}$$
   • $$\sec^2 x$$
   • $$2e^{2y}$$
   • $$4$$

9. What is the slope of the tangent line to $$\ln(xy)+y^2=x$$ at $$(e,1)$$?

   • $$\frac{e-1}{2e+1}$$
   • $$1$$
   • $$0$$
   • $$e$$

10. Find $$\frac{d^2y}{dx^2}$$ if $$x^2-2y^2=9$$.

    • $$-\frac{9}{4y^3}$$
    • $$\frac{9}{4y^3}$$
    • $$-\frac{9}{2y^3}$$
    • $$\frac{x}{2y}$$