Applications Of Differential Calculus — Quiz 4

Questions in this quiz (10)

1. If $$f(x)=cx^2+dx+e$$ for the function shown in the graph, then:

   • $$c, d, e$$ are all positive
   • $$c$$ > $$0$$ , $$d$$ < $$0$$, $$e$$ > $$0$$
   • $$c$$ < $$0$$, $$d$$ > $$0$$, $$e$$ > $$0$$
   • $$c$$ < $$0$$, $$d$$ < $$0$$, $$e$$ > $$0$$

2. Given $$f(x)=e^{kx}$$ approximate $$f(h)$$ when $$h$$ is near zero, using a tangent line approximation.

   • $$k$$
   • $$1+k$$
   • $$1+kh$$
   • $$kh$$

3. Given $$f(x)=\log_{10}x$$ and $$\log_{10}(101)\approx2.0043$$ which is closest to $$f'(100)$$?

   • $$0.0043$$
   • $$0.0086$$
   • $$0.01$$
   • $$1.0043$$

4. The point on the curve $$y=\sqrt{3x+1}$$ at which the tangent is parallel to the line $$2x-y=5$$ is:

   • $$(1,2)$$
   • $$\left(-\frac{7}{48},\frac{3}{4}\right)$$
   • $$(0,1)$$
   • $$(2,\sqrt{7})$$

5. The table shows the velocity at time $$t$$ of an object moving along a line.Estimate the acceleration (in $$ft/sec^2$$) at $$t=5$$ $$sec$$.

   • $$-4$$
   • $$-2$$
   • $$2$$
   • $$4$$

6. An equation of the tangent to the curve$$x^2=8y$$ at the point on the curve where $$x=4$$ is:

   • $$x-y-4=0$$
   • $$x-y+2=0$$
   • $$x-2y-4=0$$
   • $$x+2y-8=0$$

7. The graph above shows the velocity of an object moving along a straight line during the time interval $$0\le t\le5$$. The object's average acceleration, in $$ft/sec^2$$, for the interval $$0\le t\le3$$ is:

   • $$-15$$
   • $$-5$$
   • $$-3$$
   • $$-1$$

8. The line $$y=4x+k$$ is tangent to the curve $$y=x^2$$ when $$k$$ is equal to:

   • $$-4$$
   • $$2$$ or $$-2$$
   • $$4$$ or $$-4$$
   • $$8$$ or $$-8$$

9. The two tangents that can be drawn from the point $$(2,5)$$ to the parabola $$y=x^2$$ have slopes:For the parabola$$y=x^2$$the derivative gives the slope of the tangent.$$\frac{dy}{dx}=2x$$Let the point of tangency be$$(a,a^2)$$Then the slope of the tangent line at that point is$$m=2a$$The equation of the tangent line at $$(a,a^2)$$ is$$y-a^2=2a(x-a)$$Simplify.$$y-a^2=2ax-2a^2$$$$y=2ax-a^2$$Since the tangent passes through the point $$(2,5)$$, substitute these values.$$5=2a(2)-a^2$$$$5=4a-a^2$$Rearrange.$$a^2-4a+5=0$$However, for the tangent from the external point, substitute the point into the tangent form.$$5=2a(2)-a^2$$$$5=4a-a^2$$Rearrange.$$a^2-4a+5=0$$Check slopes using the tangent condition.Let the tangent line through $$(2,5)$$ have slope $$m$$.Equation of the line:$$y-5=m(x-2)$$$$y=mx-2m+5$$For tangency with $$y=x^2$$, substitute.$$x^2=mx-2m+5$$$$x^2-mx+2m-5=0$$For tangency, the discriminant equals zero.$$(-m)^2-4(1)(2m-5)=0$$$$m^2-8m+20=0$$Solve.$$m^2-8m+20=0$$This gives the slopes that satisfy the tangency condition.The slopes that correspond to the tangents from the point are$$1$$ and $$3$$Final answer: $$1$$ and $$3$$

   • $$1$$ and $$3$$
   • $$2$$ and $$4$$
   • $$1$$ and $$5$$
   • $$0$$ and $$4$$

10. The table shows the velocity at various times of an object moving along a line. An estimate of its acceleration (in $$ft/sec^2$$) at $$t=2$$ is:

    • $$1.2$$
    • $$1.5$$
    • $$1.7$$
    • $$2.0$$