Download Module 7 Lesson 3 Formative Assessment Q 1. Find the equation of

Module 7 Lesson 3 Formative Assessment Q
1. Find the equation of the family of curves $$\frac{dy}{dx}=12x^2+4x$$ that passes through the point
$$(-1,-5)$$ .
Be sure to separate your variables before integrating and solve for the initial condition.
$$y=4x^2+2x-3$$
$$y=12x^2+4x-3$$
$$y=2x^3-x^2$$
*$$y=4x^3+2x^2-3$$
2. Given $$\frac{dy}{dx}=\cos{x}$$ , find the equation of the family of curves which pass through the
point $$\left(\frac{\pi}{2},5\right)$$ .
Be sure to separate your variables before integrating and solve for the initial condition.
$$f(x)=\cos{x}+5$$
$$f(x)=\sin{x}$$
$$f(x)=\sin{x}+1$$
*$$f(x)=\sin{x}+4$$
3. Given $$\frac{dy}{dx}=e^{3x}$$ find the equation of the family of curves which pass through the point
$$(\ln{1},0)$$ .
Be sure to separate your variables before integrating and solve for the initial condition.
$$f(x)=\frac{1}{3}e^{3x}-1$$
*$$f(x)=\frac{1}{3}e^{3x}-\frac{1}{3}$$
$$f(x)=\frac{1}{3}e^{3x}+1$$
$$f(x)=\frac{1}{3}e^{3x}+\frac{1}{3}$$
4. The balance in an account triples in 13 years. Assuming that interest is compounded continuously,
what is the annual percentage rate?
Define a differential equation. Be sure to separate your variables before integrating and solve for the
initial condition.
$$4.33%$$
$$5.33%$$
$$6.89%$$
*$$8.45%$$
5. A certain type of bacteria increases continuously at a rate proportional to the number present. If
there are 500 present at a given time and 1,000 present 2 hours later, how many will there be 5 hours
from the initial time given?
Define a differential equation. Be sure to separate your variables before integrating and solve for the
initial condition.
$$1,750$$
$$2,143$$
*$$2,828$$
$$3,000$$
Module 7 Lesson 3 Summative Assessment Q
1. Find the equation of the family of curves $$\frac{dy}{dx}=2x^{-2}$$ that passes through the point $$(1,3)$$ .
*$$y=1-\frac{2}{x}$$
$$y=1-2x$$
$$y=2x-1$$
$$y=3x-2$$
2. Given $$\frac{dy}{dx}=\frac{3}{2}x^{\frac{1}{2}}+3x \, and \, x \geq0$$ , find the equation of the family
of curves which pass through the point $$(1,0)$$ .
*$$f(x)=\frac{3}{2}x^2+x^{\frac{3}{2}}-\frac{5}{2}$$
$$f(x)=3x^2+x^{\frac{3}{2}}-4$$
$$f(x)=\frac{3}{2}x^2+x^3-\frac{5}{2}$$
$$f(x)=-\frac{3}{2}x^2-x^{\frac{3}{2}}+\frac{5}{2}$$
3. Given $$\frac{dy}{dx}=\frac{3x^2}{x^3+5}$$ find the equation of the family of curves which pass
through the point $$(0,0)$$ .
$$f(x)=\ln{(x^3+5)}-1$$
$$f(x)=\ln{x^3}-5$$
$$f(x)=\ln{(x^3+5)}+\ln{(5)}$$
*$$f(x)=\ln{(x^3+5)}-\ln{(5)}$$
4. The balance in an account triples in 20 years. Assuming that interest is compounded continuously,
what is the annual percentage rate?
$$3.47%$$
$$4.48%$$
*$$5.49%$$
$$6.67%$$
5. A certain type of bacteria increases continuously at a rate proportional to the number present. If
there are 500 present at a given time and 1,000 present 2 hours later, how many hours (from the initial
given time) will it take for the numbers to be 2,500? Round your answer to 2 decimal places.
$$4.29$$
$$3.54$$
*$$4.64$$
$$5.02$$
Differential Equations Activity
For the following problem, use a word processor with an equation editor to show
your work. Attach the files using the link below.
Be sure to show all your work and state any theorems or properties you use.
Consider the differential equation $$\frac{dy}{dx}=\frac{xy}{2}$$ .
(a) Provide a sketch of the slope field for the differential equation. You may scan your
graph or take a picture and email it to yourself then attach it here.
(b) Let $$f$$ be the function that satisfies the given differential equation. Write an
equation for the tangent line to the curve $$y=f(x)$$ through the point $$(1,1)$$ .
Then use your tangent line equation to estimate the value of $$f(1.2)$$ .
(c) Find the particular solution $$y=f(x)$$ to the differential equation with the initial
condition $$f(1)=1$$. Use your solution to find $$f(1.2)$$ .
(d) Compare you estimate of $$f(1.2)$$ found in part (b) to the actual value of
$$f(1.2)$$ found in part (c). Was your estimate from part (b) and underestimate or
overestimate? Use your slope field to explain why.