Problem 1
Find the following: (a) \(\int 16 x^{-3} d x \quad(x \neq 0)\) (b) \(\int 9 x^{8} d x\) (c) \(\int\left(x^{5}-3 x\right) d x\) \((d) \int 2 e^{-2 x} d x\) (e) \(\int \frac{4 x}{x^{2}+1} d x\) (f) \(\int(2 a x+b)\left(a x^{2}+b x\right)^{7} d x\)
Problem 1
Evaluate the following: \((a) \int_{1}^{3} \frac{1}{2} x^{2} d x\) \((d) \int_{2}^{4}\left(x^{3}-6 x^{2}\right) d x\) \((b) \int_{0}^{1} x\left(x^{2}+6\right) d x\) \((e) \int_{-1}^{1}\left(a x^{2}+b x+c\right) d x\) \((c) \int_{1}^{3} 3 \sqrt{x} d x\) \((f) \int_{4}^{2} x^{2}\left(\frac{1}{3} x^{3}+1\right) d x\)
Problem 2
(a) Given the marginal propensity to import \(M^{\prime}(Y)=0.1\) and the information that \(M=20\) when \(Y=0,\) find the import function \(M(Y)\) (b) Civen the marginal propensity to consume \(C^{\prime}(Y)=0.8+0.1 Y^{-1 / 2}\) and the information that \(C=Y\) when \(Y=100\), find the consumption function \(C(Y)\)
Problem 2
Find: \((a) \int 13 e^{x} d x\) \((b) \int\left(3 e^{x}+\frac{4}{x}\right) d x \quad(x>0)\) \((c) \int\left(5 e^{x}+\frac{3}{x^{2}}\right) d x \quad(x \neq 0)\) \((d) \int 3 e^{-(2 x+7)} d x\) (e) \(\int 4 x e^{x^{2}+3} d x\) \((f) \int x e^{x^{2}}+9 d x\)
Problem 2
Which of the following integrals are improper, and why? \((a) \int_{0}^{x} e^{-r t} d t\) \((d) \int_{-\infty}^{0} e^{r t} d t\) (b) \(\int_{2}^{3} x^{4} d x\) \((e) \int_{1}^{5} \frac{d x}{x-2}\) (c) \(\int_{0}^{1} x^{-2 / 3} d x\) \((f) \int_{-3}^{4} 6 d x\)
Problem 3
Find: \((a) \int \frac{3 d x}{x} \quad(x \neq 0)\) \((b) \int \frac{d x}{x-2} \quad(x \neq 2)\) (c) \(\int \frac{2 x}{x^{2}+3} d x\) (d) \(\int \frac{x}{3 x^{2}+5} d x\)
Problem 3
(a) Let \(z=f(x)\) plot as a negatively sloped curve shaped like the right half of a bell in the first quadrant, passing through the points \((0,5),(2,4),(3,2),\) and (5,1) . Let \(z=g(x)\) plot as a positively sloped 45 ' line. Are \(f(x)\) and \(g(x)\) quasiconcave? (b) Now plot the sum \(f(x)+g(x)\). Is the sum function quasiconcave?
Problem 4
Given a continuous income stream at the constant rate of \(\$ 1,000\) per year: (a) What will be the present value \(\Pi\) if the income stream lasts for 2 years and the continuous discount rate is 0.05 per year? (b) What will be the present value \(\Pi\) if the income stream terminates after exactily 3 years and the discount rate is \(0.04 ?\)
Problem 4
The definite integral \(\int_{0}^{b} f(x) d x\) is said to represent an area under a curve. Does this curve refer to the graph of the integrand \(f(x)\), or of the primitive function \(F(x) ?\) If we plot the graph of the \(F(x)\) function, how can we show the given definite integral on it- -by an area, a line segment, or a point?
Problem 4
Find: (a) \(\int(x+3)(x+1)^{1 / 2} d x\) (b) \(\int x \operatorname{tr} x d x \quad(x>0)\)
