Are you ready for Math 2860?
A. Algebra
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Solve the given quadratic equations.
- (a) $r^2-4=0$
- (b) $r^2+4r-5=0$
- (c) $4r^2-5r+1=0$
- (d) $r^2-4r+4 = 0$
- (e) $r^2-5r-6 = 0$
- (f) $r^2-4r+6=0$
- (g) $r^2-6r+13=0$
- (h) $r^2-5r=0$
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Solve the given equations.
- (a) $r^4+4r^2-5=0$
- (b) $r^4-1=0$
- (c) $r^3-8=0$
- (d) $r^3-2r^2+r-2=0$
- (e) $r^3 - 3r^2+4 = 0$
- (f) $r^3-3r^2+3r-1=0$
- (g) $r^4+8r^2+16=0$
- (h) $r^4-r=0$
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Simplify the given expression.
- (a) $e^{2\ln(t)}$
- (b) $e^{-3\ln(t+1)}$
- (c) $e^{-4\ln(t)}$
- (d) $e^{3\ln(t+100)}$
- (e) $e^{t+2\ln(t)}$
B. Differentiation and Integration
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Find the first and second derivatives of the given functions. Simplify your answer.
- (a) $y=e^{3t}$
- (b) $y=x^{10}$
- (c) $y=e^{3t}\sin(2t)$
- (d) $y=te^{-2t}$
- (e) $y=e^{x^2}$
- (f) $y=e^{-2t}\cos(3t)$
- (g) $y=t^2\ln(t)$
- (h) $y=t^2e^{3t}$
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Find the integrals.
- (a) $\displaystyle\int (2x^2 + e^{2x} - \sin(5x))\,dx$
- (b) $\displaystyle\int \frac{1}{x^2+1}\,dx$
- (c) $\displaystyle\int te^{-3t}\,dt$
- (d) $\displaystyle\int \frac{3}{x}\,dx$
- (e) $\displaystyle\int \frac{1}{\sqrt{1-y^2}}\,dy$
- (f) $\displaystyle\int \frac{x}{1+x^2}\,dx$
- (g) $\displaystyle\int t\sin(t)\,dt$
- (h) $\displaystyle\int \sin(x)\tan(x)\,dx$
- (i) $\displaystyle\int \frac{1}{2t+75}\,dt$
- (j) $\displaystyle\int \frac{1}{(2t+100)^2}\,dt$
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Find the partial derivatives $u_{x}$ and $u_{y}$.
- (a) $u=y\cos x + 2xe^{y}$
- (b) $u=3x^2y+2xy+y^3$
- (c) $u=x^2\sin(3y)+xy^2$
- (d) $u=xe^{y^2}$
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Find the partial integrals.
- (a) $\displaystyle\int (y\cos x+2xe^y)\,dx$
- (b) $\displaystyle\int (y\cos(x)+2xe^y)\,dy$
- (c) $\displaystyle\int e^{2x}\sin(3y)\,dx$
- (d) $\displaystyle\int e^{2x}\sin(3y)\,dy$
Answers
A. Algebra
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Solve the given quadratic equations.
- (a) $-2; \qquad 2$
- (b) $-5; \qquad 1$
- (c) $1/4; \qquad 1$
- (d) $2; \qquad 2$
- (e) $-1; \qquad 6$
- (f) $2\pm i\sqrt{2}$
- (g) $3\pm 2i$
- (h) $0; \qquad 5$
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Solve the given equations.
- (a) $-1; \qquad 1; \qquad -i\sqrt{5}; \qquad i\sqrt{5}$
- (b) $-1; \qquad 1; \qquad -i; \qquad i$
- (c) $2; \qquad -1-i\sqrt{3}; \qquad -1+i\sqrt{3}$
- (d) $2; \qquad -i; \qquad i$
- (e) $-1; \qquad 2; \qquad 2$
- (f) $1; \qquad 1; \qquad 1$
- (g) $-2i; \qquad -2i; \qquad 2i; \qquad 2i$
- (h) $0; \qquad 1; \qquad \frac{-1\pm i\sqrt{3}}{2}$
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Simplify the given expression.
- (a) $t^2$
- (b) $(t+1)^{-3}$ or $\frac{1}{(t+1)^3}$
- (c) $t^{-4}$ or $\frac{1}{t^4}$
- (d) $(t+100)^3$
- (e) $t^2e^{t}$
B. Differentiation and Integration
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Find the first and second derivatives of the given functions. Simplify your answer.
- (a) $y' = 3e^{3t}; \qquad y'' = 9e^{3t}$
- (b) $y' = 10x^{9}; \qquad y'' = 90x^{8}$
- (c) $y' = e^{3t}(3\sin(2t) + 2\cos(2t)); \qquad y'' = e^{3t}(5\sin(2t)+12\cos(2t))$
- (d) $y' = (-2t+1)e^{-2t}; \qquad y''=(4t-4)e^{-2t}$
- (e) $y' = 2xe^{x^2}; \qquad y'' = (4x^2+2)e^{x^2}$
- (f) $y' = -e^{-2t}(3\sin(3t)+2\cos(3t)); \qquad y'' = e^{-2t}(12\sin(3t) - 5\cos(3t))$
- (g) $y' = 2t\ln(t)+t; \qquad y'' = 3+2\ln(t)$
- (h) $y' = e^{3t}(3t^2+2t); \qquad y'' = e^{3t}(9t^2+12t+2)$
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Find the integrals.
- (a) $\frac{2}{3}x^3 + \frac{1}{2}e^{2x}+\frac{1}{5}\cos(5x)+C$
- (b) $\arctan(x)+C$
- (c) $-\frac{1}{3}te^{-3t}-\frac{1}{9}e^{-3t}+C$
- (d) $3\ln|x|+C$
- (e) $\arcsin(y)+C$
- (f) $\frac{1}{2}\ln(x^2+1)+C$
- (g) $\sin(t)-t\cos(t)+C$
- (h) $-\sin(x)+\ln|\sec(x)+\tan(x)|+C$
- (i) $\frac{1}{2}\ln|2t+75|+C$
- (j) $-\frac{1}{2}\cdot\frac{1}{2t+100}+C$
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Find the partial derivatives $u_{x}$ and $u_{y}$.
- (a) $u_{x} = -y\sin x + 2e^{y}; \qquad u_{y} = \cos x +2xe^{y}$
- (b) $u_{x} = 6xy+2y; \qquad u_{y} = 3x^2+2x+3y^2$
- (c) $u_{x} = 2x\sin(3y)+y^2; \qquad u_{y} = 3x^2\cos(3y)+2xy$
- (d) $u_{x} = e^{y^2}; \qquad u_{y} = 2xye^{y^2}$
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Find the partial integrals.
- (a) $y\sin x + x^2e^{y}+h(y)$, where $h(y)$ does not depend on $x$.
- (b) $\frac{1}{2}y^2\cos(x)+2xe^{y}+h(x)$, where $h(x)$ does not depend on $y$.
- (c) $\frac{1}{2}e^{2x}\sin(3y)+h(y)$
- (d) $-\frac{1}{3}e^{2x}\cos(3y)+h(x)$
