Free Printable Calc II - Midterm 1 Review

1.

Find f⁻¹(3) where f(x) = x³ + 2x

a)

0

b)

3

c)

1

d)

-1

2.

Find (f⁻¹)'(3) where f(x) = x³ + 2x

a)

5

b)

1/5

c)

-5

d)

-1/5

3.

Find the inverse function $f(x)=\frac{(x-1)}{(x+1)}$

a)

$f^{-1}(x)=\frac{(x+1)}{(x-1)}$

b)

$f^{-1}(x)=\frac{(1-x)}{(1+x)}$

c)

$f^{-1}\left(x\right)=\frac{\ \left(x+1\right)}{\left(1-x\right)}$

d)

$f^{-1}\left(x\right)=\frac{\ \left(x+1\right)}{\left(1+x\right)}$

4.

Use log differentiation to find the derivative of $y=\frac{(x-1)^{\frac{1}{5}}}{(x^2+1)^{\frac{1}{2}}}$

a)

$y'=\frac{(x-1)^{-\frac{4}{5}}}{(x^2+1)^{\frac{1}{2}}}-\frac{x(x-1)^{\frac{1}{5}}}{(x^2+1)^{\frac{3}{2}}}$

b)

$y'=\frac{1}{5}\frac{(x-1)^{\frac{1}{5}}}{(x^2+1)^{\frac{1}{2}}}-\frac{x(x-1)^{\frac{1}{5}}}{(x^2+1)^{\frac{3}{2}}}$

c)

$y'=\frac{1}{5}\frac{(x-1)^{-\frac{4}{5}}}{(x^2+1)^{\frac{1}{2}}}-\frac{x(x-1)^{\frac{1}{5}}}{(x^2+1)^{\frac{3}{2}}}$

d)

$y'=\frac{1}{5}\frac{(x-1)^{-\frac{4}{5}}}{(x^2+1)^{\frac{1}{2}}}-\frac{x(x-1)^{\frac{1}{5}}}{(x^2+1)^{\frac{1}{2}}}$

5.

Differentiate with respect to x:

$x^{\pi}e^{(x^2+1)}$

a)

$\pi x^{(\pi-1)}e^{(x^2+1)}-2x^{1+\pi}e^{(x^2+1)}$

b)

$\pi x^{(\pi-1)}e^{(x^2+1)}+2x^{1+\pi}e^{(x^2+1)}$

c)

$\pi x^{(\pi-1)}e^{(x^2+1)}$

d)

$2x^{1+\pi}e^{(x^2+1)}$

6.

Differentiate with respect to x:

$\ln(x^2+3)$

a)

$\frac{x}{(x^2+3)}$

b)

$\frac{1}{\left(x^2+3\right)}$

c)

$\frac{2}{(x^2+3)}$

d)

$\frac{2x}{(x^2+3)}$

7.

Differentiate with respect to x:

$\left(\ln(x^3+1)\right)^{e^x}$

a)

$\frac{e^{e^x\ln\left(\ln\left(x^3+1\right)\right)}}{\ln\left(x^3+1\right)}\ \frac{3x^2}{x^3+1}e^{x\ }$ $+\ln\left(\ln\left(x^3+1\right)\right)e^xe^{e^x\ln\left(\ln\left(x^3+1\right)\right)}$

b)

$\frac{e^x\left(\ln\left(x^3+1\right)\right)^{e^x-1}}{\left(x^3+1\right)}3x^2$

c)

$\frac{e^{\ln\left(x^3+1\right)e^x}3x^2e^x}{x^3+1}$

d)

$\frac{e^{\ln\left(x^3+1\right)e^x}\ 3x^2}{\left(x^3+1\right)}e^{x\ }$

$+e^{\ln\left(x^3+1\right)e^x}\ln\left(x^3+1\right)e^x$

8.

$\int\ \cos\left(x\right)e^{\sin\left(x\right)+1}\ dx$

a)

$e^u+C$

b)

$e^{\sin\left(x\right)}+C$

c)

$e^{\sin\left(x\right)+1}+C$

d)

$e^{\sin\left(x\right)+C}$

9.

Integrate: $\int\frac{e^x}{(e^x+1)}dx$

a)

$\frac{e^x}{(e^x+1)}+C$

b)

$\ln(e^x+1)+C$

c)

$\ln\left|e^x-1\right|+C$

d)

$e^x+C$

10.

Integrate: $\int\ \frac{1}{\sqrt{4-25x^2}}\ dx$

a)

$\frac{\ 1}{5}\sin^{-1}\left(\frac{5x}{2}\right)\ +C$

b)

$\frac{\ 1}{5}\tan^{-1}\left(\frac{5x}{2}\right)\ +C$

c)

$\frac{\ 1}{5}\sec^{-1}\left(\frac{5x}{2}\right)\ +C$

d)

$\frac{\ 1}{2}\sin^{-1}\left(\frac{2x}{5}\right)\ +C$

11.

John deposited $500 in a bank account. How much money will John have after 2 years with 5% interest compounded continuously?

a)

$500e^{\left(0.05\right)2}$ dollars

b)

$500e^{5\cdot2}$ dollars

c)

$500\left(0.05\right)2$ dollars

d)

$0.05e^{500\cdot2}$ dollars

12.

Which of the following is an integrating factor for the differential equation

$t\frac{\ dy}{dt}+2y=t^2-t+1$

a)

$e^{\int\frac{2}{t}dt}$

b)

$e^{\int2dt}$

c)

$e^{\int2tdt\ }$

d)

$e^{\int tdt}$

13.

Use the general solution

$y(t)=\frac{1}{4}t^2−\frac{1}{3}t+\frac{1}{2}+\frac{c}{t^2}$

of the previous differential equation to solve the initial value problem $y\left(1\right)=\frac{1}{2}$ .

a)

$y(t)=\frac{1}{4}t^2−\frac{1}{3}t+\frac{1}{2}+\frac{7}{t^2}$

b)

$y(t)=\frac{1}{4}t^2−\frac{1}{3}t+\frac{1}{2}+\frac{1}{12t^2}$

c)

$y(t)=\frac{1}{4}t^2−\frac{1}{3}t+\frac{1}{2}+\frac{12}{t^2}$

d)

$y(t)=\frac{1}{4}t^2−\frac{1}{3}t+\frac{1}{2}+\frac{1}{7t^2}$

Calc II - Midterm 1 Review

Total questions: 13

Worksheet time: 22mins

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