Chapter 6

(a) Find the derivatives of \(\sin \left(x^{2}+1\right)\) and \(\sin \left(x^{3}+1\right)\) (b) Use your answer to part (a) to find antiderivatives of: (i) \(x \cos \left(x^{2}+1\right)\) (ii) \(x^{2} \cos \left(x^{3}+1\right)\) (c) Find the general antiderivatives of: (i) \(x \sin \left(x^{2}+1\right)\) (ii) \(x^{2} \sin \left(x^{3}+1\right)\)

Using the Fundamental Theorem, evaluate the definite integrals in Problems \(1-20\) exactly. $$\int_{0}^{4} 6 x d x$$

Find the integrals. $$\int t e^{5 t} d t$$

Decide if the function is an antiderivative of \(f(x)=2 e^{2 x}\) $$F(x)=e^{2 x}+5$$

Suppose \(F^{\prime}(x)=2 x^{2}+5\) and \(F(0)=3 .\) Find the value of \(F(b)\) for \(b=0,0.1,0.2,0.5,\) and 1.0.

Explain how you can tell if substitution can be used to find an antiderivative. $$\int x\left(1-5 x^{2}\right)^{5} d x$$

Using the Fundamental Theorem, evaluate the definite integrals in Problems \(1-20\) exactly. $$\int_{1}^{3} 5 d x$$

Find the integrals. $$\int p e^{-0.1 p} d p$$

Find the present and future values of an income stream of \(\$ 12,000\) a year for 20 years. The interest rate is \(6 \%\) compounded continuously.

Decide if the function is an antiderivative of \(f(x)=2 e^{2 x}\) $$F(x)=e^{2 x}$$

Explain how you can tell if substitution can be used to find an antiderivative. $$\int \frac{\sqrt{\ln x}}{x} d x$$

Using the Fundamental Theorem, evaluate the definite integrals in Problems \(1-20\) exactly. $$\int_{0}^{3} t^{3} d t$$

Find the integrals. $$\int y \ln y \, d y$$

Find the present and future values of an income stream of \(\$ 2,000\) per year for 15 years, assuming a \(5 \%\) interest rate compounded continuously.

Decide if the function is an antiderivative of \(f(x)=2 e^{2 x}\) $$F(x)=5 e^{2 x}$$

Explain how you can tell if substitution can be used to find an antiderivative. $$\int \frac{x}{\sqrt{\ln x}} d x$$

Using the Fundamental Theorem, evaluate the definite integrals in Problems \(1-20\) exactly. $$\int_{0}^{2}\left(12 x^{2}+1\right) d x$$

Find the integrals. $$\int(z+1) e^{2 z} d z$$

Assuming an interest rate of \(3 \%\) compounded continuously, (a) Find the future value in 10 years of a payment of \(\$ 10,000\) made today. (b) Find the future value of an income stream of \(\$ 1000\) per year over 10 years. (c) Which is larger, the future value from the lump sum in part (a) or from the income stream in part (b)? Explain why this makes sense financially.

Decide if the function is an antiderivative of \(f(x)=2 e^{2 x}\) $$F(x)=x e^{2 x}$$

Given the demand curve \(p=35-q^{2}\) and the supply curve \(p=3+q^{2},\) find the producer surplus when the market is in equilibrium.

Explain how you can tell if substitution can be used to find an antiderivative. $$\int \sin ^{9} t \cos t d t$$

Using the Fundamental Theorem, evaluate the definite integrals in Problems \(1-20\) exactly. $$\int_{0}^{2}\left(3 t^{2}+4 t+3\right) d t$$

Find the integrals. $$\int q^{5} \ln 5 q d q$$

Assuming an interest rate of \(5 \%\) compounded continuously, (a) Find the future value in 6 years of a payment of \(\$ 12,000\) made today. (b) Find the future value of an income stream of \(\$ 2000\) per year over 6 years. (c) Which is larger, the future value from the lump sum in part (a) or from the income stream in part (b)? Explain why this makes sense financially.

Find the consumer surplus for the demand curve \(p=\) \(100-4 q\) when \(q^{*}=10\) items are sold at the equilibrium price.

Find the integrals .Check your answers by differentiation. $$\int 2 x\left(x^{2}+1\right)^{5} d x$$

Using the Fundamental Theorem, evaluate the definite integrals in Problems \(1-20\) exactly. $$\int_{1}^{2} \frac{1}{x} d x$$

Find the integrals. $$\int y \sqrt{y+3} d y$$

A person deposits money into an account, which pays \(5 \%\) interest compounded continuously, at a rate of \(\$ 1000\) per year for 15 years. Calculate: (a) The balance in the account at the end of the 15 years. (b) The amount of money actually deposited into the account. (c) The interest earned during the 15 years.

Decide if the function is an antiderivative of \(f(x)=2 e^{2 x}\) $$F(x)=e^{2 x}+\int_{0}^{1} e^{2 t} d t$$

Find the integrals .Check your answers by differentiation. $$\int \frac{x}{\sqrt{x^{2}+4}} d x$$

Using the Fundamental Theorem, evaluate the definite integrals in Problems \(1-20\) exactly. $$\int_{1}^{4} \frac{1}{\sqrt{x}} d x$$

Find the integrals. $$\int x^{3} \ln x d x$$

A person deposits money into an account, which pays \(6 \%\) interest compounded continuously, at a rate of \(\$ 1000\) per year for 30 years. Calculate: (a) The balance in the account at the end of the 30 years. (b) The amount of money actually deposited into the account. (c) The interest earned during the 30 years.

Decide whether the expression is a number or a family of functions. (Assume \(f(x)\) is a function.) $$5+\int 2 f(x) d x$$

The demand curve for a product is given by \(q=100-2 p\) and the supply curve is given by \(q=3 p-50\) (a) Find the consumer surplus at the equilibrium. (b) Find the producer surplus at the equilibrium.

Find the integrals .Check your answers by differentiation. $$\int(5 x-7)^{10} d x$$

Using the Fundamental Theorem, evaluate the definite integrals in Problems \(1-20\) exactly. $$\int_{0}^{1}\left(6 q^{2}+4\right) d q$$

Find the integrals. $$\int(t+2) \sqrt{2+3 t} d t$$

(a) Find the present and future value of an income stream of \(\$ 6000\) per year for a period of 10 years if the interest rate, compounded continuously, is \(5 \%\) (b) How much of the future value is from the income stream? How much is from interest?

Decide whether the expression is a number or a family of functions. (Assume \(f(x)\) is a function.) $$\int_{3}^{5} 2 f(x) d x$$

For a product, the demand curve is \(p=100 e^{-0.008 q}\) and the supply curve is \(p=4 \sqrt{q}+10\) for \(0 \leq q \leq 500\) where \(q\) is quantity and \(p\) is price in dollars per unit. (a) At a price of \(\$ 50,\) what quantity are consumers willing to buy and what quantity are producers willing to supply? Will the market push prices up or down? (b) Find the equilibrium price and quantity. Does your answer to part (a) support the observation that market forces tend to push prices closer to the equilibrium price? (c) At the equilibrium price, calculate and interpret the consumer and producer surplus.

Find the integrals .Check your answers by differentiation. $$\int x \sqrt{x^{2}+1} d x$$

Using the Fundamental Theorem, evaluate the definite integrals in Problems \(1-20\) exactly. $$\int_{0}^{5} 3 x^{2} d x$$

Find the integrals. $$\int \frac{y}{\sqrt{5-y}} d y$$

A small business expects an income stream of \(\$ 5000\) per year for a four- year period. (a) Find the present value of the business if the annual interest rate, compounded continuously, is (i) \(3 \%\) (ii) \(10 \%\) (b) In each case, find the value of the business at the end of the four-year period.

Decide whether the expression is a number or a family of functions. (Assume \(f(x)\) is a function.) $$\int 5 \, d x$$

Sketch possible supply and demand curves where the consumer surplus at the equilibrium price is (a) Greater than the producer surplus. (b) Less than the producer surplus.

Find the integrals .Check your answers by differentiation. $$\int 2 q e^{q^{2}+1} d q$$