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Mathematics LibreTexts

3.5E: Exercises for Section 3.5

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In exercises 1 - 10, find dydx for the given functions.

1) y=x2secx+1

Answer
dydx=2xsecxtanx

2) y=3cscx+5x

3) y=x2cotx

Answer
dydx=2xcotxx2csc2x

4) y=xx3sinx

5) y=secxx

Answer
dydx=xsecxtanxsecxx2

6) y=sinxtanx

7) y=(x+cosx)(1sinx)

Answer
dydx=(1sinx)(1sinx)cosx(x+cosx)

8) y=tanx1secx

9) y=1cotx1+cotx

Answer
dydx=2csc2x(1+cotx)2

10) y=(cosx)(1+cscx)

In exercises 11 - 16, find the equation of the tangent line to each of the given functions at the indicated values of x. Then use a calculator to graph both the function and the tangent line to ensure the equation for the tangent line is correct.

11) [T] f(x)=sinx,x=0

Answer

y=x

The graph shows negative sin(x) and the straight line T(x) with slope −1 and y intercept 0.

12) [T] f(x)=cscx,x=π2

13) [T] f(x)=1+cosx,x=3π2

Answer

y=x+23π2

The graph shows the cosine function shifted up one and has the straight line T(x) with slope 1 and y intercept (2 – 3π)/2.

14) [T] f(x)=secx,x=π4

15) [T] f(x)=x2tanx,x=0

Answer

y=x

The graph shows the function as starting at (−1, 3), decreasing to the origin, continuing to slowly decrease to about (1, −0.5), at which point it decreases very quickly.

16) [T] f(x)=5cotx,x=π4

In exercises 17 - 22, find d2ydx2 for the given functions.

17) y=xsinxcosx

Answer
d2ydx2=3cosxxsinx

18) y=sinxcosx

19) y=x12sinx

Answer
d2ydx2=12sinx

20) y=1x+tanx

21) y=2cscx

Answer
d2ydx2=2csc(x)(csc2(x)+cot2(x))

22) y=sec2x

23) Find all x values on the graph of f(x)=3sinxcosx where the tangent line is horizontal.

Answer
x=(2n+1)π4,where n is an integer

24) Find all x values on the graph of f(x)=x2cosx for 0<x<2π where the tangent line has slope 2.

25) Let f(x)=cotx. Determine the points on the graph of f for 0<x<2π where the tangent line(s) is (are) parallel to the line y=2x.

Answer
(π4,1),(3π4,1),(5π4,1),(7π4,1)

26) [T] A mass on a spring bounces up and down in simple harmonic motion, modeled by the function s(t)=6cost where s is measured in inches and t is measured in seconds. Find the rate at which the spring is oscillating at t=5 s.

27) Let the position of a swinging pendulum in simple harmonic motion be given by s(t)=acost+bsint. Find the constants a and b such that when the velocity is 3 cm/s, s=0 and t=0.

Answer
a=0,b=3

28) After a diver jumps off a diving board, the edge of the board oscillates with position given by s(t)=5cost cm at t seconds after the jump.

a. Sketch one period of the position function for t0.

b. Find the velocity function.

c. Sketch one period of the velocity function for t0.

d. Determine the times when the velocity is 0 over one period.

e. Find the acceleration function.

f. Sketch one period of the acceleration function for t0.

29) The number of hamburgers sold at a fast-food restaurant in Pasadena, California, is given by y=10+5sinx where y is the number of hamburgers sold and x represents the number of hours after the restaurant opened at 11 a.m. until 11 p.m., when the store closes. Find y and determine the intervals where the number of burgers being sold is increasing.

Answer
y=5cos(x), increasing on (0,π2),(3π2,5π2), and (7π2,12)

30) [T] The amount of rainfall per month in Phoenix, Arizona, can be approximated by y(t)=0.5+0.3cost, where t is months since January. Find yand use a calculator to determine the intervals where the amount of rain falling is decreasing.

For exercises 31 - 33, use the quotient rule to derive the given equations.

31) ddx(cotx)=csc2x

32) ddx(secx)=secxtanx

33) ddx(cscx)=cscxcotx

34) Use the definition of derivative and the identity cos(x+h)=cosxcoshsinxsinh to prove that ddx(cosx)=sinx.

For exercises 35 - 39, find the requested higher-order derivative for the given functions.

35) d3ydx3 of y=3cosx

Answer
d3ydx3=3sinx

36) d2ydx2 of y=3sinx+x2cosx

37) d4ydx4 of y=5cosx

Answer
d4ydx4=5cosx

38) d2ydx2 of y=secx+cotx

39) d3ydx3 of y=x10secx

Answer
d3ydx3=720x75tan(x)sec3(x)tan3(x)sec(x)

3.5E: Exercises for Section 3.5 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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